Quadratic Goldreich-Levin Theorems

Quadratic Goldreic h-Levin Theorems Madh ur T ulsiani ∗ Julia W olf † No v em b er 5, 2018 Abstract Decomp osition theorems in classical F ourier analysis enable us to express a b ounded function in terms of few linear phases with la rge F ourier co eﬃcie nts plus a part that is pseudorandom with resp ect to linear phases . The Goldreich-Levin a lg orithm [GL89] can b e viewed a s an alg o rithmic analogue o f such a deco mpo sition as it gives a wa y to eﬃcie ntly ﬁnd the linea r phas es asso cia ted with large F ourier co eﬃcients. In the s tudy of “ quadratic F ourier a nalysis”, hig he r -degree analogue s of such dec o mpo sitions hav e been developed in which t he pseudor andomness proper ty is stronger but the structured part corr e sp ondingly weaker. F or example, it has previo us ly b een shown that it is p oss ible to express a b ounded function as a sum of a few qua dratic phases plus a par t that is small in the U 3 norm, deﬁned b y Gow ers fo r the purp o se of counting ar ithmetic progressio ns of leng th 4 . W e give a polyno mial time algorithm for computing such a decompositio n. A key part of the a lgorithm is a lo cal self-co r rection pro cedure for Reed-Muller co des of order 2 (ov er F n 2 ) for a function at distance 1 / 2 − ε from a co deword. Given a function f : F n 2 → {− 1 , 1 } at fra ctional Hamming distance 1 / 2 − ε from a quadra tic phase (whic h is a co deword of Reed- Muller co de of order 2), we g ive an algo rithm that runs in time p olynomia l in n and ﬁnds a co deword at distance at most 1 / 2 − η for η = η ( ε ). This is an algorithmic analogue of Samoro dnitsky’s result [Sam0 7], whic h gave a tester for the a bove problem. T o our knowledge, it repr esents the ﬁrs t instance of a correction pro cedure for any class of co des, b eyond the list-deco ding radius. In the pro c e ss, we give alg orithmic v er sions of results from additive combinatorics used in Samoro dnitsky’s pr o of and a reﬁned version of the in verse theorem for the Gow er s U 3 norm ov er F n 2 . ∗ Princeton U niversit y and IAS, Princeton, NJ. W ork supp orted b y NSF grant CCF-0832797. † Cen tre de Math´ ematiques Lauren t Sc hw artz, ´ Ecole Polytec hniq ue, 91128 P alaiseau, F rance. 1 In tro du c tion Higher-order F ourier analysis, whic h has its ro ots in Go wers’s pro of of Szemer´ edi’s T h eorem [Go w98], has exp erienced a signiﬁcan t surge in the n umb er of a v ailable to ols as well as applications in recen t years, includin g p erhap s most notably Gree n and T ao’s pro of that there are arbitrarily long arithmetic progressions in the primes. Across a range of mathematica l d isciplin es, classical F ourier analysis is often applied in form of a de c omp osition the or em : one writes a b ound ed function f as f = f 1 + f 2 , (1) where f 1 is a structured part consisting of the frequencies with large amplitud e, while f 2 consists of the remainin g frequencies and resem bles uniform, or r andom-lo oking, noise. Over F n 2 , the F ourier basis consists of functions of the form ( − 1) h α,x i for α ∈ F n 2 , w hic h w e shall refer to as line ar phase functions . The p art f 1 is then a (weig hted) sum of a few linear phase fu nctions. F rom an algorithmic p oint of view, eﬃcien t tec h niques are av ailable to compute the structured part f 1 . The Goldreich-Levin [GL89] theorem giv es an algorithm wh ic h computes, with high p robabilit y , the large F ourier co eﬃcien ts of f : F n 2 → {− 1 , 1 } in time p olynomial in n . One w a y of viewing this theorem is pr ecisely as an algo rith m ic version of the decomp osition th eorem ab o ve , where f 1 is the part consisting of large F ourier co eﬃcients of a function and f 2 is random-lo oking with resp ect to an y test that can only detect large F ourier coeﬃcient s. It w as observed b y Go we rs (and previously by F ursten b erg and W eiss in the context of ergod ic theory) that the coun t of certain patterns is not almost in v ariant under the addition o f a noise term f 2 as deﬁn ed ab o ve , and th us a deco mp osition su c h as (1) is not su ﬃcien t in that con text. In particular, for coun ting 4-te rm arithmetic progressions a more sensitiv e notion of uniformity is needed. T his subtler notion of uniformit y , called quadr atic uniformity , is expressed in terms of the U 3 norm, whic h w as in tro duced by Go wers in [Go w98 ] and whic h w e shall deﬁne b elo w. In certain situations we ma y therefore wish to d ecomp ose the function f as ab o v e, but w h ere the random-lo oking part is quadratically uniform, meaning k f 2 k U 3 is small. Naturally one needs to answ er th e question as to what replaces the structur e d p art , whic h in (1) w as deﬁ ned b y a small n umb er of linear c haracters. This questio n b elongs to the realm of what is now called quadr atic F ourier analysis . Its ce ntral building b lo c k, largely cont ained in Gow ers ’s pro of o f Szemer ´ edi’s theorem bu t reﬁned by Green and T ao [GT08] and S amoro dnitsky [Sam07], is the so-called inverse the or em for the U 3 norm , whic h state s, roughly sp eaking, that a fun ction with large U 3 norm correlates with a quadr atic phase function , by which we mean a function of the form ( − 1) q for a qu adratic form q : F n 2 → F 2 . The in v erse theorem implies that th e str uctured part f 1 has quadratic structure in the case wh ere f 2 is s mall in U 3 , and starting with [Gre07] a v ariet y of suc h quadr atic de c omp osition the or ems hav e come in to existence: in one formulat ion [GW10c], one can w r ite f as f = X i λ i ( − 1) q i + f 2 + h, (2) where the q i are quadr atic forms, th e λ i are real coeﬃcient s such that P i | λ i | is b ounded, k f 2 k U 3 is small and h is a sm all ℓ 1 error (that is negligible in all kno wn applications.) In analogy with th e d ecomp osition in to F ourier charact ers, it is n atural to think of the coeﬃcient s λ i as the quadr atic F ourier c o eﬃcients of f . As in the case of F ourier coeﬃcien ts, there is a trade- oﬀ b et w een the complexit y of the structured part and the randomn ess of the uniform part. In 1 the case of the quad r atic decomp osition ab o ve, the b ound on the ℓ 1 norm of the coeﬃcients λ i dep end s in v ersely on the uniform it y parameter k f 2 k U 3 . Ho wev er, unlik e the decomp osition into F ourier c haracters, the decomp osition in terms of quadratic phases is not necessarily u nique, as the quadratic phases do not form a b asis for the space of functions on F n 2 . Quadratic decomposition theorems ha v e found sev er al num b er-theoretic applications, notably in a series of pap ers by Go w ers and t he sec ond author [GW10c , GW10a, GW10b], as w ell as [Can10 ] and [HL11]. Ho w eve r, all decomp osition theorems of this typ e prov ed so f ar hav e b een of a rather abstract nature. In particular, work b y T revisan, V adh an and the ﬁrst author [TTV09] uses linear programming tec hniques and b o osting, while Go w ers and the s econd author [GW10c ] ga ve a (non-constru ctiv e) existence pro of using the Hahn-Banac h theorem. The b o osting p ro of is constructive in a v ery w eak sense (see Section 3) but is quite far fr om giving an algorithm for computing the ab ov e decomp ositions. W e giv e suc h an algo r ith m in this pap er. A computer science p ersp ective. Algorithmic decomp osition theorems, su c h as the weak regularit y lemma of F rieze and K annan [FK99] w hic h decomp oses a matrix as a small sum of cut matrices, h av e found numerous app lication in appro ximately solving constrain t satisfactio n problems. F rom th e p oint of view of theoretical computer science, a v ery natural question to ask is if the simple description of a bou n ded f unction as a small list of qu adratic phases can b e computed eﬃcien tly . In this pap er we give a probabilistic algorithm that p erforms th is task, us in g a n u mb er of reﬁnements of ingredien ts in th e proof of the inv erse theorem to mak e it more e ﬃcient, wh ic h will b e d etailed b elo w. Connections to Reed-Muller co des. A building block in pro ving the d ecomp osition theorem is an alg orithm for the follo win g problem: giv en a function f : F n 2 → {− 1 , 1 } , whic h is at Hamming distance at most 1 / 2 − ε from an u nknown quadr atic phase ( − 1) q , ﬁn d ( eﬃcient ly) a quadratic phase ( − 1) q ′ whic h is at d istance at most 1 / 2 − η from f , for some η = η ( ε ). This naturally leads to a connection w ith Reed-Muller co d es since for Reed-Muller co des of order 2, the codewords are precisely the (truth-tables of ) quadratic phases. Note that the list deco ding radiu s of Reed-Muller co d es of order 2 is 1 / 4 [GKZ08, Gop10], whic h means that if the distance we re l ess t h an 1 / 4, w e could ﬁnd al l such q , and th er e w ould only be p oly( n ) man y of t h em. Th e distance here is greater than 1 / 4 and there migh t b e exp onen tially man y (in n ) suc h functions q . Ho wev er, the problem may still b e tractable as we are required to ﬁnd only one such q (whic h migh t b e at a sligh tly larger distance than q ′ ). The prob lem o f testing if th ere is suc h a q was considered b y Samoro dnitsky [Sam07]. W e sho w that in fact, the result can b e turn ed into a lo c al self c orr e ctor for Reed-Muller co des at distance (1 / 2 − ε ). W e are not a w are of an y class of co des for w h ic h such a self-co rr ecting pro cedu re is kno wn, b eyond the list-decod in g radius. 1.1 Ov erview of results and techniqu es W e state b elo w the basic decomp osition theorem for quadr atic ph ases, whic h is obtained by com- bining Theorems 3.1 and 4.1 pro v ed later. Th e theorem is stated in term s of the U 3 norm, d eﬁned formally in Sectio n 2. Theorem 1.1 L et ε, δ > 0 , n ∈ N and B > 1 . Then ther e exists η = exp(( B /ε ) C ) and a r andomize d algorithm running in time O ( n 4 log n · p oly( η , log(1 /δ ))) which, given any function 2 g : X → [ − 1 , 1] as an or acle, outputs with pr ob ability at le ast 1 − δ a de c omp osition i nto quadr atic phases g = c 1 ( − 1) q 1 + . . . + c k ( − 1) q k + e + f satisfying k ≤ 1 /η 2 , k f k U 3 ≤ ε , k e k 1 ≤ 1 / 2 B and | c i | ≤ η for al l i . Note that in [GW10 a ] the authors had to w ork muc h harder to obtain a b ound on the num b er of terms in the decomp osition, rather than just t h e ℓ 1 norm o f its coeﬃcient s. Our decomp osition approac h give s suc h a b ound immediately and is equiv alen t fr om a quan titativ e p oint of view: w e can b ound the num b er of terms here b y 1 /η 2 , w hic h is exp onenti al in 1 /ε . It is p ossible to f u rther strengthen this theorem by com bining the quadratic p h ases obtained into only p oly(1 /ε ) quadr atic aver ages . Roughly sp eaking, ea ch qu adratic a v erage is a sum of few quadratic phases, whic h diﬀer only in their linear part. W e describ e this in detail in Section 5. The k ey comp onent of the ab ov e d ecomp osition theorem is the follo win g self-correction pro cedure for Reed-Muller cod es of order 2 (whic h are simply truth-tables of quadratic phase functions). The correlation b et we en tw o functions f and g is deﬁned as h f , g i = E x ∈ F n 2 [ f ( x ) g ( x ) ]. Theorem 1.2 Given ε, δ > 0 , ther e exists η = exp( − 1 /ε C ) and a r andomize d algorithm Find-Qua dratic r unning in time O ( n 4 log n · p oly(1 /ε, 1 /η, log (1 /δ ))) which, g iven or acle ac c ess to a function f : F n 2 → {− 1 , 1 } , either outputs a quadr atic form q ( x ) or ⊥ . The algorithm satisﬁes the fol lowing guar ante e. • If k f k U 3 ≥ ε , then with pr ob ability at le ast 1 − δ it ﬁnds a quadr atic form q such that h f , ( − 1) q i ≥ η . • The pr ob ability that the algorithm outputs a quadr atic form q with h f , ( − 1) q i ≤ η / 2 is at most δ . W e remark that all the resu lts conta ined here can b e extended to F n p for an y constan t p . W e c ho ose to presen t only the case of F n 2 for simplicit y of notation. Our results f or computing the ab o v e d ecomp ositions comprise v arious comp onents. Constructiv e decomposition theorems. W e pro v e the decomp osition theorem us ing a pro ce- dure which, at ev ery step, tests if a certain function has co rr elation at least 1 / 2 − ε with a quadratic phase. Giv en an algorithm to ﬁnd such a q u adratic ph ase, the pro cedure gives a w a y to com bin e them to obtain a decomp osition. Previous decomp osition theorems ha v e also used su c h p ro cedures [FK99, TTV09]. Ho wev er, th ey required that the quadratic phase found at eac h step hav e corr elation η = O ( ε ), if one exists with correlation ε . In particular, they require the fact that if w e scale f to change its ℓ ∞ norm, the quan tities η and ε would scale the same w a y (this w ould not b e true if, sa y , η = ε 2 ). W e need and pr o v e a general decomp osition th eorem, whic h w orks ev en as η d egrades arbitrarily in 1 /ε . This requires a somewhat more sophisticated analysis and the intro d uction of a third error term for whic h w e b ound the ℓ 1 norm. Algorithmic versions of theorems from additive com bina t orics. Samoro dn itsky’s pro of uses sev eral results f rom additiv e com b in atorics, which pro duce large sets in F n 2 with certain usefu l additiv e prop erties. The pro of of the inv erse theorem uses the descrip tion of these sets. Ho w eve r, 3 in our setting, we d o not hav e time to lo ok at the en tire set since they m a y b e of size p oly( ε ) · 2 n , as in the case of the Balog-Szemer ´ edi-Go wers theo r em describ ed later. W e th us w ork by bu ilding eﬃcien t sampling pro cedures or pro cedures f or eﬃcien tly deciding memb ership in such sets, wh ic h require new alg orithmic p ro ofs. A subtlet y arises when one tries to construct suc h a testing pro cedu re. Since the pro cedure runs in p olynomial time, it often works b y sampling and estimating certain prop erties and the estimates ma y b e erroneous. T h is leads to some noise in the decision of an y suc h an algorithm, resulting a noisy version of the set (actually a distribution o ver sets). W e get around this pr oblem b y pro ving a robust version of the Balog-Sze mer´ edi-Go w ers theorem, for whic h w e can “sand wic h” the output of suc h a pr o cedure b et wee n t wo sets with desirable prop erties. This tec hnique ma y be useful in other algorithmic applicatio ns. Lo cal in v erse theorems and decompositions inv olving quadratic a v erages. Samoro d nit- sky’s in ve rse t h eorem sa ys that wh en a fun ction f has U 3 norm ε , then one can ﬁ nd a quadr atic phase q wh ic h has correlatio n η w ith f , for η = exp( − 1 /ε C ). A decomp osition then r equires 1 /η 2 , that is exp onen tially many (in 1 /ε ), terms. A s omewhat s tr onger r esult was implicit in the w ork of Green and T ao [GT08]. They s h o we d that there exists a su bspace of co dimension p oly(1 /ε ) and on all of wh ose cosets f correlates p olynomially with a quadratic ph ase. Picking a p articular coset and extending th at quadr atic ph ase to th e whole space giv es the pr evious theorem. It turn s out that the diﬀeren t qu adratic p hases on eac h coset in fact ha v e the same quadratic part and d iﬀer only by a linear term. Th is wa s exploited in [GW10c ] to obtain a decomp osition in vol vin g only p olynomially man y quadratic ob jects, so-called quadr atic aver ages , w hic h are describ ed in more detail in Sectio n 5. W e remark that the results of Green and T ao [GT08] do not directly extend to the case of c harac- teristic 2 since division by 2 is used at one crucial p oin t in the argument. W e combine their ideas with those of Samoro dnitsky to giv e an algorithmic v ersion of a decomp osition th eorem in volvi n g quadratic a v erages. 2 Preliminaries Throughout the p ap er, we shall b e using L atin lette rs such as x , y or z to denote elemen ts of F n 2 , while Greek letters α and β are used to denote mem b ers of the dual space c F n 2 ∼ = F n 2 . W e s hall use δ as our error p arameter, wh ile ε, η , γ and ρ are v ariously u sed to indicate correlation strength b et ween a Bo olean fun ction f and a family of stru ctured fun ctions Q . Thr oughout the man uscript N will denote the quan tit y 2 n . Constants C ma y c hange from line to lin e without further notice. W e shall b e using the follo wing standard probabilistic b oun ds without further men tion. Lemma 2.1 (Ho eﬀding bound for sampling [TV06]) If X is a r andom variable with | X | ≤ 1 and ˆ µ is the empiric al aver age obtaine d fr om t samples, then P [ | E [ X ] − ˆ µ | > γ ] ≤ exp( − Ω( γ 2 t )) . A Hoeﬀdin g-t yp e b ound can also b e obtained for p olynomial functions of ± 1-v alued rand om v ari- ables. 4 Lemma 2.2 (Ho eﬀding bound for low-degree p olynomials [O’D08 ]) Supp ose that F = F ( X 1 , . . . , X N ) is a p olynomial of de gr e e d in r andom variables X 1 , . . . , X N taking value ± 1 , then P [ | F − E [ F ] | > γ ] ≤ exp  − Ω  d · ( γ /σ ) 2 /d  , wher e σ = q E [ F 2 ] − E [ F ] 2 is the standar d d evi ation of F . W e start oﬀ by stating t wo fundament al r esu lts in additiv e combinatorics which are often applied in sequence. F or a set A ⊆ F n 2 , w e w r ite A + A f or the set of elemen ts a + a ′ suc h that a, a ′ ∈ A . More generally , the k -fold sumset , denoted b y k A , consists of all k -fold su ms of element s of A . First, the Balog-Szemer ´ edi-Go wers theorem states that if a set has many additiv e quadruples, that is, elemen ts a 1 , a 2 , a 3 , a 4 suc h that a 1 + a 2 = a 3 + a 4 , then a large subs et of it m ust ha ve small sumset. Theorem 2.3 (Balog-Szemer´ edi-Go wers [Go w98]) L et A ⊆ F n 2 c ontain at le ast | A | 3 /K addi- tive quadrup les. Then ther e exists a subset A ′ ⊆ A of size | A ′ | ≥ K − C | A | with the pr op erty that | A ′ + A ′ | ≤ K C | A ′ | . F reiman’s theorem, ﬁr st pro v ed by Ru zsa in the context of F n 2 , asserts that a set with s m all sumset is eﬃcie ntly con tained in a subspace. Theorem 2.4 (F reiman-Ruzsa Theorem [R uz99]) L et A ⊆ F n 2 b e such that | A + A | ≤ K | A | . Then A is c ontaine d in a sub sp ac e of size at most 2 O ( K C ) | A | . W e shall also require the notion of a F r eiman homomorp hism . W e sa y th e map l is a F reiman 2-homomorphism if x + y = z + w imp lies l ( x ) + l ( y ) = l ( z ) + l ( w ). More ge ner ally , a F r eiman homomorphism of order k is a map l suc h that x 1 + x 2 + · · · + x k = x ′ 1 + x ′ 2 + · · · + x ′ k implies that l ( x 1 ) + · · · + l ( x k ) = l ( x ′ 1 ) + · · · + l ( x ′ k ). Th e order of the F reiman homomorphism measures the degree of linearit y of l ; in particular, a truly linear m ap is a F reiman homomorphism of all orders. Next w e recall the deﬁn ition of the uniformity of U k norms in tro du ced by Go wers in [Go w98 ]. Deﬁnition 2.5 L et G b e any ﬁnite ab elian gr oup. F or any p ositive inte ger k ≥ 2 and any function f : G → C , deﬁne the U k -norm by the formula k f k 2 k U k = E x,h 1 ,...,h k ∈ G Y ω ∈{ 0 , 1 } k C | ω | f ( x + ω · h ) , wher e ω · h is shorthand for P i ω i h i , and C | ω | f = f if P i ω i is ev e n and f otherwise. In the sp ecial case k = 2, a compu tation sho ws that k f k U 2 = k b f k l 4 , and hence any approac h using the U 2 norm is essentially equiv alen t to using ord inary F ourier analysis. I n the case k = 3, the U 3 norm counts the n umber of additiv e octuples “co ntained in” f , that is, we a verage ov er the pro duct of f at all eigh t v ertices of a 3-dimensional parallelepip ed in G . 5 These uniformit y norms satisfy a num b er of imp ortant prop erties: they are clearly nested k f k U 2 ≤ k f k U 3 ≤ k f k U 4 ≤ ... and can b e deﬁned ind uctiv ely k f k 2 k +1 U k +1 = E x k f x k 2 k U k , where k ≥ 2 and th e function f x stands for the assignmen t f x ( y ) = f ( y ) f ( x + y ). Thin king of th e function f as a complex exp onent ial (a phase function), we can int erp r et th e fun ction f x as a kind of discr ete derivative o f f . It follo ws straight from a s im p le bu t admittedly ingenious sequence of ap p lications of the Cauc h y- Sc hw arz inequalit y that if the balanced fu nction 1 A − α of a set A ⊆ G of densit y α has small U k norm, th en A con tains the expected n um b er of arithmetic progressions of length k + 1, namely α k +1 | G | 2 . Th is fact make s the un iformit y norms in teresting for n umber -theoretic applications. In computer scie n ce they h a v e b een u sed in the cont ext of p r obabilistically c hec k ab le pro ofs (PCP) [ST06], comm un ication complexit y [VW 07], as w ell as in the analysis of pseudo-rand om generators that fool lo w-degree p olynomials [BV10]. In man y app lications, b eing small in the U k norm is a desirable pr op erty for a fu nction to ha v e. What can w e say if this is not the case? It is not to o diﬃcult to verify th at k f k U k = 1 if and only if f is a p olynomial ph ase function of degree k − 1, i.e. a function of the form ω p ( x ) where p is a p olynomial of degree k − 1 a n d ω is an appropriate ro ot o f unity . But d o es every function with large U k norm look like a p olynomial phase function of d egree k − 1? It turns out th at any function w ith large U k norm correlat es, at the very least lo cally , with a p olynomial phase fun ction of degree k − 1. This is kno wn as the inv er s e th eorem for th e U k norm, pro ved b y Green and T ao [GT08] for k = 3 and p > 2 and Samoro d nitsky [Sam07] for k = 3 and p = 2, and Bergelson, T ao and Z iegler [BTZ10, TZ10] for k > 3. W e shall r estrict our atten tion to the case k = 3 in this pap er, whic h we can state as follo ws . Theorem 2.6 (Global Inv erse Theorem for U 3 [GT08], [Sam07]) L et f : F n p → C b e a function such that k f k ∞ ≤ 1 and k f k U 3 ≥ ε . Then ther e e xists a a quadr atic form q and a ve ctor b such that | E x f ( x ) ω q ( x )+ b · x | ≥ exp ( − O ( ε − C )) In Sectio n 5 w e shall d iscuss v arious reﬁnemen ts of the in ve rse th eorem, including correlations with so-calle d quadr atic aver ages . These reﬁ nemen ts allo w us to obtain p olynomial instead of exp onen tial correlation with some qu ad r atically structured ob ject. W e discuss f urther p oten tial impro v ements and extensions of the argumen ts presented in th is pap er in Section 6 . First of all, how ev er, w e shall turn to the problem of constructiv ely obtaining a decomp osition assuming that one h as an eﬃcien t correlati on testing p r o cedure, whic h is done in Section 3. 3 F rom d ecomp ositions to correlation testing In t h is sec tion we redu ce f rom th e pr ob lem of ﬁn ding a decomp osition f or giv en function to t h e problem of ﬁ n ding a single quadratic phase or av erage th at correlat es w ell with the fu n ction. 6 W e state the basic decomp osition result in somewhat greater generalit y as we b eliev e it ma y b e of indep en d en t in terest. W e will consider a real-v alued function g on a ﬁn ite domain X (whic h shall b e F n 2 in the r est of the pap er). W e shall decomp ose the function g in terms of members from an arbitrary class Q of fun ctions q : X → [ − 1 , 1]. Q ma y later b e tak en to b e the class of quadr atic phases or quadr atic a v erages. W e will assume Q to b e closed un der negatio n of the fu nctions i.e., q ∈ Q ⇒ − q ∈ Q . Finally , we shall consider a semi-norm k·k S deﬁned for functions on X , suc h that if k f k S is large for f : X → R then f has large correlation w ith some fu n ction in Q . The ob vious c hoice for k·k S is k f k S = max q ∈Q |h f , q i| , as is the case in man y kno wn decomposition r esults and the general result in [TTV09]. Ho wev er, w e will b e able to obtai n a stronger alg orithmic guaran tee b y taking k·k S to b e the U 3 norm. Theorem 3.1 L et Q b e a class of functions as ab ove and let ε, δ > 0 and B > 1 . L e t A b e an algorithm which, given or acle ac c ess to a function f : X → [ − B , B ] satisfying k f k S ≥ ε , outputs, with pr ob ability at le ast 1 − δ , a fu nc tion q ∈ Q such tha t h f , q i ≥ η for some η = η ( ε, B ) . Then ther e exists an algo rithm which, given any function g : X → [ − 1 , 1] , outputs with pr ob ability at le ast 1 − δ /η 2 a de c omp osition g = c 1 q 1 + . . . + c k q k + e + f satisfying k ≤ 1 /η 2 , k f k S ≤ ε and k e k 1 ≤ 1 / 2 B . Also, the algorithm makes at most k c al ls to A . W e pro v e the decomp osition theorem building on an argument from [TT V09], wh ic h in turn gen- eralizes an argum ent of [FK99]. Both the arguments in [TTV09, FK99] work well if for a f u nction f : X → R satisfying max q ∈Q | h f , q i | ≥ ε , one can eﬃcien tly ﬁnd a q ∈ Q with h f , q i ≥ η = Ω( ε ). It is imp ortan t there that η = Ω( ε ), or at least that the guarant ee is indep en d en t of h o w f is scale d. Both pr o ofs giv e an algorithm w hic h, at eac h s tep t , c hecks if there exists q t ∈ Q whic h has goo d correlation with a giv en function f t , an d the decomp osition is ob tained by adding the f u nctions q t obtained at diﬀerent steps. In b oth cases, the ℓ ∞ norm of the fu nctions f t c hanges as the algo rith m pro ceeds. Supp ose ε ′ = o ( ε ) and we only had the scale-dep end en t guaran tee that for fun ctions f : X → [ − 1 , 1] with k f k S ≥ ε , we can eﬃcien tly ﬁnd a q ∈ Q suc h that h f , q i ≥ ε 2 (sa y). Then at step t of the algorithm if w e hav e k f t k ∞ = M (sa y), then k f t k S ≥ ε will imply k f / M k S ≥ ε/ M and one can only ge t a q t satisfying h f t , q t i ≥ M · ( ε/ M ) 2 = ε 2 / M . Thus, the correlation of the functions q t w e can obtain d egrades as th e k f t k ∞ increases. T his turn s out to b e ins uﬃcien t to b ound the num b er of steps r equired b y these algorithms and hence the n umb er of terms in the decomp osition. When testing correlations with qu adratic phases usin g k·k S as the U 3 norm, the correlatio n η obtained for f : F n 2 → [ − 1 , 1] has v ery bad dep end ence on ε and hence w e ru n into the ab o ve problem. T o get aroun d it, we tru ncate the functions f t used b y the algorithm so that w e ha v e a uniform b ound on their ℓ ∞ norms. How ever, this truncation in tro d u ces an extra term in the decomp osition, for whic h w e b oun d the ℓ 1 norm. Con trolling the ℓ 1 norm of th is term requires a somewhat more soph isticated analysis than in [FK99]. An analysis based on a similar p otentia l function w as also emplo yed in [TTV09] (though not for the p urp ose of con trolling the ℓ 1 norm). W e n ote that a third term with b ounded ℓ 1 norm also app ears in th e (non-constructiv e) decomp o- sitions obtained in [GW10a]. Pro of of T heorem 3.1: W e will assum e a ll calls to the algorithm A correctly return a q as ab o ve or d eclare k f k S < ε as th e case ma y b e. Th e probabilit y of an y err or in the calls to A is at most k δ . 7 W e build the decomp osition b y the follo wing simple pr o cedure. - Deﬁne functions f 1 = h 1 = g . Set t = 1. - While k f t k S ≥ ε – Let q t b e the output of A when call ed with th e f unction f t . – h t +1 := h t − η q t . – f t +1 := Truncate [ − B ,B ] ( h t +1 ) = max {− B , min { B , h t +1 }} – t := t + 1 If the algorithm ru ns f or k steps, the decomposition it outputs is g = k X t =1 η · q t + ( h k − f k ) + f k where w e tak e f = f k and e = h k − f k . By construction, we hav e th at k f k k S ≤ ε . It remains to sho w that k ≤ 1 /η 2 and k h k − f k k 1 ≤ 1 / 2 B . T o analyze k h t − f t k , w e will deﬁne an additional function ∆ t def = f t · ( h t − f t ). Note that ∆ t ( x ) ≥ 0 for every x , since f t is simply a truncation of h t and hence f t = B when h t > f t and − B when h t < f t . This giv es k ∆ t k 1 = E [∆ t ] = E [ f t · ( f t − h t )] = E [ B · | h t − f t | ] = B · k h t − f t k 1 . W e will in fact boun d th e ℓ 1 norm of ∆ k to obtain the required b ound on k h k − f k k 1 . The follo wing lemma states the b ounds we need at ev ery step. Lemma 3.2 F or eve ry input x and every t ≤ k − 1 f 2 t ( x ) − f 2 t +1 ( x ) + 2∆ t ( x ) − 2∆ t +1 ( x ) + η 2 ≥ 2 η · q t ( x ) f t ( x ) . W e ﬁr st sho w ho w the ab o ve lemma suﬃces to p ro ve the theorem. T aking expectations on b oth sides of the inequalit y gives, for all t ≤ k − 1, k f t k 2 2 − k f t +1 k 2 2 + 2 k ∆ t k 1 − 2 k ∆ t +1 k 1 + η 2 ≥ 2 η · h q t , f t i ≥ 2 η 2 . Summing o ver all t ≤ k − 1 giv es k f 1 k 2 2 − k f k k 2 2 + 2 k ∆ 1 k 1 − 2 k ∆ k k 1 ≥ k · η 2 = ⇒ k · η 2 + k f k k 2 2 + 2 k ∆ k k 1 ≤ 1 since k f 1 k 2 2 = k g k 2 2 ≤ 1 and ∆ 1 = 0. Ho w ev er, this giv es k ≤ 1 /η 2 and k ∆ k k 1 ≤ 1 / 2, whic h in turn implies k h k − f k k 1 ≤ 1 / 2 B , completing the pro of of Theorem 3.1. W e no w return to th e pro of of Lemma 3.2. Pro of of Lemma 3.2: W e shall ﬁx an input x and consider all functions only at x . W e start b y bringing the RHS into the desired form and collecting terms. 2 η q t · f t = 2( h t − h t +1 ) · f t = 2( h t − f t ) · f t − 2( h t +1 − f t +1 ) · f t +1 + 2 f 2 t − 2 f 2 t +1 − 2 h t +1 · f t + 2 h t +1 · f t +1 = 2∆ t − 2∆ t +1 + f 2 t − f 2 t +1 +  f 2 t − f 2 t +1 − 2 h t +1 ( f t − f t +1 )  8 It remains to show that f 2 t − f 2 t +1 − 2 h t +1 ( f t − f t +1 ) = ( f t − f t +1 )( f t + f t +1 − 2 h t +1 ) ≤ η 2 . W e ﬁrst note that if | f t +1 | < B , then h t +1 = f t +1 and the expression b ecomes ( f t − f t +1 ) 2 , which is at most η 2 . Also, if | f t | = | f t +1 | = B , then f t and f t +1 m ust b e equal (as f t only c hanges in steps of η ) and the expression is 0. Finally , in the case w hen | f t | < B and | f t +1 | = B , we must ha v e that | f t − h t +1 | = | h t − h t +1 | ≤ η . W e can then b ound the expr ession as ( f t − f t +1 )( f t + f t +1 − 2 h t +1 ) ≤  ( f t − f t +1 ) + ( f t + f t +1 − 2 h t +1 ) 2  2 = ( f t − h t +1 ) 2 ≤ η 2 , whic h pro ves the lemma. W e next sho w that in the case when k·k S is the U 3 norm and Q contai n s at most exp ( o (2 n )) functions, it is suﬃcien t to test the correla tions only for Boolean functions f : F n 2 → {− 1 , 1 } . This can be d one by simply scal ing a f unction taking v alues in [ − B , B ] to [ − 1 , 1] a nd then randomly rounding the v alue indep end ently at eac h input to ± 1 with appropriate p robabilit y . Lemma 3.3 L et ε, ¿ . 0 . L e t A b e an algorithm, which, given or acle ac c ess to a function f : F n 2 → {− 1 , 1 } satisfying k f k U 3 ≥ ε , outputs, with pr ob ability at le ast 1 − δ , a function q ∈ Q such tha t h f , q i ≥ η for som e η = η ( ε ) . In add ition, assume tha t the running time of A is p oly( n, 1 /η, log (1 /δ )) . Then ther e exists an algorithm A ′ which, given or acle ac c ess to a function f : F n 2 → [ − B , B ] satisfying k f k U 3 ≥ ε , outputs, with pr ob ability at le ast 1 − 2 δ , an element q ∈ Q satisfying h f , q i ≥ η ′ for η ′ = η ′ ( ε, B ) . M or e over, the running time of A ′ is p oly( n, 1 /η ′ , log (1 /δ )) . Pro of: Consider a rand om Bo olean function ˜ f : F n 2 → {− 1 , 1 } s uc h that ˜ f ( x ) is 1 with probabilit y (1 + f ( x ) /B ) / 2 and − 1 otherwise. A ′ simply calls A with the f unction ˜ f an d parameters ε/ 2 B , δ . This means that whenever A queries the v alue of the function at x , A ′ generates it indep endently of all other p oints by lo oking at f ( x ). It then outp uts the q giv en by A . If k ˜ f k U 3 ≥ ε/ 2 B , then A outpu ts a q satisfying h ˜ f , q i ≥ η ( ε/ 2 B ). If for th e same q w e also ha ve h f , q i ≥ B · η ( ε/ 2 B ) / 2 = η ′ ( ε, B ), then the output of A ′ is as desired. Ho wev er, k ˜ f k U 3 is a p olynomial of degree 8 and the correlatio n with an y q is a linear p olynomial in the 2 n random v ariables { ˜ f ( x ) } x ∈ F n 2 . Th u s , b y Lemma 2.2, the prob ab ility that k ˜ f k U 3 < k f k U 3 /B − ε/ 2 B , or h ˜ f , q i ≥ h f , q i /B − η ( ε/ 2 B ) / 2 for an y q ∈ Q , is at most exp ( − Ω ε,B ( −|Q| · 2 n )) ≤ δ . Th u s, to compute the requir ed decomp osition in to qu adratic p hases, one only needs to giv e an algorithm for ﬁndin g a phase q = ( − 1) q satisfying h f , ( − 1) q i ≥ η when f : F n 2 → {− 1 , 1 } is a Bo ole an function sati sfy in g k f k U 3 ≥ ε . 4 Finding correlated quadratic p h ases o ve r F n 2 In this section, w e sho w ho w to obtain an alg orithm for ﬁ nding a quadratic phase wh ic h has goo d correlation with a giv en function Boolean f : F n 2 → {− 1 , 1 } (if one exists). F or an f satisfying k f k U 3 ≥ ε , w e w ant to ﬁnd a quadratic form q su c h that h f , ( − 1) q i ≥ η ( ε ). The follo wing th eorem pro vides suc h a guaran tee. 9 Theorem 4.1 Given ε, δ > 0 , ther e exists η = exp( − 1 /ε C ) and a r andomize d algorithm Find-Qua dratic r unning in time O ( n 4 log n · p oly(1 /ε, 1 /η, log (1 /δ ))) which, g iven or acle ac c ess to a fu nction f : F n 2 → {− 1 , 1 } , either outputs a quadr atic phase ( − 1) q ( x ) or ⊥ . The algor ithm satisﬁes the fol lowing guar ante e. • If k f k U 3 ≥ ε , then with pr ob ability at le ast 1 − δ it ﬁnds a quadr atic form q such that h f , ( − 1) q i ≥ η . • The pr ob ability that the algorithm outputs a quadr atic form q with h f , ( − 1) q i ≤ η / 2 is at most δ . The fact that k f k U 3 ≥ ε implies the existenc e of a quadratic phase ( − 1) q with h f , ( − 1) q i ≥ η w as pro ven b y Samorod nitsky [Sam07]. W e giv e an algorithmic version of his pro of, starting with the pro ofs of the r esults from add itiv e com binatorics con tained therein. Note that k f k 8 U 3 is simply the exp ected v alue of the pro du ct Q ω ∈{ 0 , 1 } 3 f ( x + ω · h ) for random x, h 1 , h 2 , h 3 ∈ F n 2 . Hence, Lemma 2.1 implies th at k f k U 3 can b e easily estimate d by sampling suﬃcien tly many v alues of x, h 1 , h 2 , h 3 and taking th e a v erage of the pro du cts for the samples. Corollary 4.2 By making O ((1 /γ 2 ) · log (1 /δ )) queries to f , one c an obtain an estimate ˆ U such that P h | k f k U 3 − ˆ U | > γ i ≤ δ. The main algorithm b egins b y chec king if ˆ U ≥ 3 ε/ 4 and rejects if this is not the case. If ˆ U ≥ 3 ε/ 4, then the ab o ve claim implies that k f k U 3 ≥ ε/ 2 with high p robabilit y . So our algorithm will actually return a q with correlation η ( ε ′ ) with ε ′ = ε/ 2. W e s hall ignore this and just use ε in the sequel for the sak e of readabilit y . 4.1 Pic king large F ourier co eﬃcien ts in deriv atives The ﬁrst step of the p ro of in [Sam07] is to ﬁn d a c hoice function ϕ : F n 2 → F n 2 whic h is “somewhat linear”. The c hoice function is u sed to pick a F ourier coeﬃcient for the deriv ativ e f y . The in tuition is that if f w ere indeed a qu adratic phase of the form ( − 1) h x,M x i , then f y ( x ) = f ( x ) f ( x + y ) = ( − 1) h x, ( M + M T ) y i · ( − 1) h y, M y i . Th u s, the largest F ourier coeﬃcien t (with absolute v alue 1) w ould b e ˆ f y (( M + M T ) y ). Hence, there is a function ϕ ( y ) def = ( M + M T ) y , whic h is given by m ultiplying y by a symm etric matrix M + M T , whic h selects a large F ou r ier co eﬃcien t for f y . The pr o of att emp ts to construct suc h a sym metric matrix for an y f with k f k U 3 ≥ ε . Expanding the U 3 norm an d usin g H¨ older’s inequalit y giv es the follo wing lemma. Lemma 4.3 (Corollary 6.6 [Sam07 ]) Supp ose that f : F n 2 → {− 1 , 1 } is such tha t k f k U 3 ≥ ε . Then E x,y   X α,β ˆ f x 2 ( α ) · ˆ f y 2 ( β ) · d f x + y 2 ( α + β )   ≥ ε 16 . 10 Cho osing a random fun ction ϕ ( x ) = α with probabilit y ˆ f x 2 ( α ) satisﬁes P x,y [ ϕ ( x ) + ϕ ( y ) = ϕ ( x + y )] = X α,β ˆ f x 2 ( α ) · ˆ f y 2 ( β ) · d f x + y 2 ( α + β ) . Th u s, when k f k U 3 ≥ ε , the abov e lemma giv es that P ϕ,x,y [ ϕ ( x ) + ϕ ( y ) = ϕ ( x + y )] = E x,y   X α,β ˆ f x 2 ( α ) · ˆ f y 2 ( β ) · d f x + y 2 ( α + β )   ≥ ε 16 . The p ro of in [Sam07] w orks with a random f unction ϕ as describ ed ab o v e. W e deﬁne a slight ly diﬀeren t random f unction ϕ , s in ce w e need its v alue at any input x to b e samplable in time p olynomial in n . Thus, w e will only sample α for which th e corresp onding F our ier co eﬃcien ts are suﬃcien tly large. In particular, w e need an algo rithmic v ersion of the decomp osition of a fu nction in to linear phases, whic h follo ws from the Goldreic h-Levin th eorem. Theorem 4.4 (Goldreic h-Levin [GL89]) L et γ , δ > 0 . Ther e is a r andomize d algorithm Linear-D ecomposi tion , which, given or acle ac c e ss to a function f : F n 2 → {− 1 , 1 } , runs in time O ( n 2 log n · p oly(1 /γ , log (1 /δ ))) and outputs a de c omp osition f = k X i =1 c i · ( − 1) h α i ,x i + f ′ with the fol lowing guar ante e: • k = O (1 /γ 2 ) . • P h ∃ i | c i − ˆ f ( α i ) | > γ / 2 i ≤ δ . • P h ∀ α such that | ˆ f ( α ) | ≥ γ , ∃ i α i = α i ≥ 1 − δ . Remark 4.5 Note that the ab ove is a slightly non-standa r d version of the Goldr eich-L evin the or em. The usual one makes O ( n log n · p oly (1 /γ , log (1 /δ ))) queries to f (wher e e ach query takes O ( n ) time to write down) and guar ante es that for any sp e ciﬁc α such that | ˆ f ( α ) | ≥ γ , ther e exists an i with α i = α , with pr ob ability at le ast 1 − δ . By r ep e ating the algorithm O (log (1 /γ )) times, we c an take a union b ound over al l α as in the last pr op erty guar ante e d by the ab ove the or em. It follo ws that in order to sample ϕ ( x ), ins tead of samp ling fr om all F ourier co eﬃcien ts of f x , we only sample from the large F our ier coeﬃcien ts usin g the ab o v e decomp osition. W e shall denote the quan tit y ε 16 / 4 that app ears b elo w b y ρ . Lemma 4.6 Ther e e xi sts a distribution over func tions ϕ : F n 2 → F n 2 such that ϕ ( x ) is indep endently chosen for e ach x ∈ F n 2 , and is samplable in time O ( n 3 log n · p oly(1 /ε )) given or acle ac c ess to f . Mor e over, if k f k U 3 ≥ ε , then we have P ϕ  P x,y [ ϕ ( x ) + ϕ ( y ) = ϕ ( x + y )] ≥ ε 16 / 4  ≥ ε 16 / 4 . 11 Pro of: W e sample ϕ ( x ) at eac h inpu t x as follo ws. W e run L inear-De composit ion for f x with γ = δ = ε 16 / 18 and sample ϕ ( x ) to b e α i with probabilit y c 2 i . If P c 2 i < 1, we answer arbitrarily with the remaining p robabilit y . By Theorem 4.4, with pr ob ab ility at least 1 − 2 γ o ver the run of Linear -Decompo sition , eac h α ∈ F n 2 with | ˆ f x ( α ) | ≥ γ is samp led with probabilit y at least ( ˆ f x ( α ) − γ / 2) 2 ≥ ˆ f x 2 ( α ) − γ . Let [ z ] 0 denote max { 0 , z } . W e hav e P ϕ,x,y [ ϕ ( x ) + ϕ ( y ) = ϕ ( x + y )] ≥ E x,y   X α,β (1 − 2 γ ) 3 h ˆ f x 2 ( α ) − γ i 0 h ˆ f y 2 ( β ) − γ i 0 h d f x + y 2 ( α + β ) − γ i 0   ≥ ε 16 − 9 γ , whic h by our choic e of parameters is at least ε 16 / 2. This immediately implies that P ϕ  P x,y [ ϕ ( x ) + ϕ ( y ) = ϕ ( x + y )] ≥ ε 16 / 4  ≥ ε 16 / 4. Th u s, with probabilit y ρ = ε 16 / 4 one gets a goo d ϕ whic h is somewhat linear. This ϕ is then used to reco ver an appropriate qu adratic phase. W e will actually dela y sampling the function on all p oint s and only query ϕ ( x ) when needed in the construction of the quadr atic phase (whic h we sho w can b e done b y querying ϕ o n p olynomially man y p oin ts). Consequently , the construction pro cedur es that follo w will only work w ith a sm all prob ab ility , i.e. w hen we are actually working with a goo d ϕ . How ever, we can te st the quadratic phase w e obtain in the en d and rep eat the en tire p ro cess if th e ph ase do es not correlate w ell with f . Also, note that we store the ( x, ϕ ( x )) already sampled in a data structure and r e-use them if and when the same x is queried again. 4.2 Applying the Balog-S zemer´ edi-Go w ers theorem The next step of th e pro of uses ϕ to obtain a linear c hoice function D x for some matrix D . This step uses certain results from additiv e com b inatorics, for whic h w e develo p alg orithmic v ersions b elo w. In particular, it applies the Balog-Szemer ´ edi-Go w ers (BSG) theorem to the set A ϕ def = n ( x, ϕ ( x )) : | ˆ f x ( ϕ ( x )) | ≥ γ o , where w e will choose γ = O ( ε 16 ) as in Lemma 4.6. F or any set A ∈ { 0 , 1 } n that is somewhat linear, th e Balog- Szemer´ edi-Go w ers theorem allo ws u s to ﬁnd a subset A ′ ⊆ A whic h is large and do es not gro w too m u ch w hen added to itself. W e state the follo w ing v ersion from [BS94], whic h is particularly su ited to our application. Theorem 4.7 (Balog-Szemer´ edi-Go wers Theorem [BS94]) L e t A ⊆ F n 2 b e su c h that P a 1 ,a 2 ∈ A [ a 1 + a 2 ∈ A ] ≥ ρ . Then ther e exists A ′ ⊆ A , | A | ′ ≥ ρ | A | such that | A ′ + A ′ | ≤ (2 /ρ ) 8 | A | . W e are in terested in ﬁnd ing the set A ′ ϕ whic h results from app lying the ab ov e theorem to the set A ϕ . Ho wev er, since the set A ′ ϕ is of expon ential size, we do not ha v e time to write do w n the en tire set (ev en if w e can ﬁnd it). In stead, w e w ill need an eﬃcien t algorithm for testing mem b ership in the set. T o get the required algorithmic version, w e follo w the pro of b y Sud ak o v, Sze mer´ edi and V u [SSV05] and the pr esen tation b y Viola [Vio 07 ]. In this pro of one actually constru cts a graph on the s et A ϕ and then selects a sub set of the neigh b orho o d of a ran d om v er tex as A ′ ϕ , after remo ving certain pr oblematic v ertices. It can b e deduced that the set A ′ ϕ can b e found in time p olynomial in the size of the graph. Ho w eve r, as 12 discussed ab o v e, this is still exp onential in n a nd hence inadequ ate for our purp oses. Belo w, w e dev elop a test to c heck if a certain elemen t ( x, ϕ ( x )) is in A ′ ϕ . W e ﬁrst d eﬁne a (random) graph on the v ertex set 1 { ( x, ϕ ( x )) | x ∈ F n 2 } and edge set E γ for γ > 0, deﬁned as E γ def =    ( x, ϕ ( x )) , ( y , ϕ ( y ))       ϕ ( x ) + ϕ ( y ) = ϕ ( x + y ) and | ˆ f x ( ϕ ( x )) | , | ˆ f y ( ϕ ( y )) | , | d f x + y ( ϕ ( x + y )) | ≥ γ    . Lemma 4.6 implies that o v er the c hoice of ϕ , with p r obabilit y at least ρ = ε 16 / 4, the graph deﬁned with γ = ε 16 / 18, has densit y at least ρ . How ever, if a ϕ is go o d for a certain v alue of γ , then it is also go o d for all v alues γ ′ ≤ γ (as the density of the grap h can only increase). F or the remaining argumen t, w e will assume that w e hav e s amp led ϕ completely and that i t is g o o d. W e will later c ho ose γ ∈ [ ε 16 / 180 , ε 16 / 18]. Since we will b e examining the prop erties of certai n n eighb orh o o ds in this graph, we ﬁrst w rite a pro cedur e to test if t wo v ertices in the graph ha ve an edge b et ween th em. Edge-Tes t (u,v, γ ) - Let u = ( x, ϕ ( x )) and v = ( y , ϕ ( y )). - Estimate | ˆ f x ( ϕ ( x )) | , | ˆ f y ( ϕ ( y )) | and | d f x + y ( ϕ ( x + y )) | usin g t samples for eac h. - Answer 1 if ϕ ( x ) + ϕ ( y ) = ϕ ( x + y ) and all estimates are at lea st γ , and 0 otherwise. Unfortunately , s in ce we are only estimating the F our ier coeﬃcients, w e will only b e able to test if t wo ve rtices hav e an edge b et ween them with a sligh t error in the th reshold γ , and with high probabilit y . Th u s, if the estimate is at lea st γ , we ca n only sa y th at with high probabilit y , the F ourier coeﬃcient m ust b e at least γ − γ ′ for a s m all error γ ′ . This leads to the follo wing guarant ee on Edge-Tes t . Claim 4.8 Given γ ′ , δ > 0 , the output of E dge-Test ( u, v , γ ) with t = O (1 /γ ′ 2 · log (1 /δ )) qu eries, satisﬁes the fol lowing guar ante e with pr ob ability at le ast 1 − δ . • Edge-Tes t ( u, v , γ ) = 1 = ⇒ ( u, v ) ∈ E γ − γ ′ . • Edge-Tes t ( u, v , γ ) = 0 = ⇒ ( u, v ) / ∈ E γ + γ ′ . Pro of: The claim follo ws immed iately from Lemma 2.1 and the deﬁnitions of E γ − γ ′ , E γ + γ ′ . The approximat e nature of the ab o ve test in tro du ces a subtle issue. Note that the outputs 1 and 0 of th e test corresp ond to th e p resence or abs ence of edges in diﬀer ent gr aphs with ed ge sets E γ − γ ′ and E γ + γ ′ . The edge sets of the tw o graphs are r elated as E γ + γ ′ ⊆ E γ − γ ′ . But the pro of of Theorem 4.7 uses somewh at m ore complicat ed subsets of v ertices, wh ic h are d eﬁned using b oth upp er and lo w er b ounds on th e sizes of certain n eigh b orh o o ds. Since the u pp er and lo w er b oun ds estimated using the ab o ve test will h old for slightly diﬀeren t graphs, we need to b e careful in analyzi n g any algorithm that u ses Edge-Tes t as a p rimitiv e. 1 Since ϕ is rand om, the ver tex set of the graph as deﬁned is random. Ho wev er, since ϕ is a fun ction, the vertex set is isomorphic to F n 2 and one ma y think of the graph as b eing deﬁned on a ﬁxed set of vertices with edges c hosen according to a rand om pro cess. 13 W e no w r eturn to the argument as pr esen ted in [SS V05]. It considers the neighborh o o d of a r an d om v ertex u a n d remo v es vertice s that ha ve too few neigh b ors in common with other v ertices in the graph. L et the size of the verte x set b e N = 2 n . F or a v ertex u , w e deﬁne the follo wing s ets: N ( u ) def = { v : ( u, v ) ∈ E γ } S ( u ) def =  v ∈ N ( u ) : P v 1  v 1 ∈ N ( u ) and | N ( v ) ∩ N ( v 1 ) | ≤ ρ 3 N  ≥ ρ 2  =  v ∈ N ( u ) : P v 1  v 1 ∈ N ( u ) and P v 2 [ v 2 ∈ N ( v ) ∩ N ( v 1 )] ≤ ρ 3  > ρ 2  T ( u ) def = N ( u ) \ S ( u ) =  v ∈ N ( u ) : P v 1  v 1 ∈ N ( u ) and P v 2 [ v 2 ∈ N ( v ) ∩ N ( v 1 )] ≤ ρ 3  ≤ ρ 2  It is sho wn in [SSV05] (see also [Vio07]) that if the graph has densit y ρ , th en p ic king A ′ ϕ = T ( u ) for a r andom vertex u is a go o d c hoice 2 . Lemma 4.9 L et the gr aph with e dge set E γ have density at le ast ρ and let A ′ ϕ = T ( u ) for a r andom vertex u . Then, with pr ob ability at le ast ρ/ 2 over the choic e of u , the set A ′ ϕ satisﬁes   A ′ ϕ   ≥ ρN and   A ′ ϕ + A ′ ϕ   ≤ (2 /ρ ) 8 N . W e no w translate the condition for mem b ership in the set T ( u ) into an algorithm. Note that w e p erform diﬀeren t edge tests with diﬀerent th resholds, the v alues of whic h will b e c hosen later. BSG-Test ( u, v , γ 1 , γ 2 , γ 3 , ρ 1 , ρ 2 ) (Appro ximate test to c hec k if v ∈ T ( u )) - Let u = ( x, ϕ ( x )) and v = ( y , ϕ ( y )). - Sample ( z 1 , ϕ ( z 1 )) , . . . , ( z r , ϕ ( z r )). - F or eac h i ∈ [ r ], sample ( w ( i ) 1 , ϕ ( w ( i ) 1 )) , . . . , ( w ( i ) s , ϕ ( w ( i ) s )). - If Edge-T est (u,v, γ 1 ) = 0, then output 0. - F or i ∈ [ r ] , j ∈ [ s ], let X i = Edge-T est ( ( x, ϕ ( x )) , ( z i , ϕ ( z i )) , γ 2 ) Y ij = Edge-T est  ( y , ϕ ( y )) ,  w ( i ) j , ϕ  w ( i ) j  , γ 3  Z ij = Edge-T est  ( z i , ϕ ( z i )) ,  w ( i ) j , ϕ  w ( i ) j  , γ 3  - F or eac h i , tak e B i = 1 if 1 s P j Y ij · Z ij ≤ ρ 1 and 0 ot h er w ise. - Answer 1 if 1 r P i X i · B i ≤ ρ 2 and 0 otherwise. 2 Note that here we are c ho osing A ′ ϕ to b e th e neighborho od of any vertex in the graph, instead of vertices in A ϕ . How ever, th is is not a p roblem since the on ly ve rtices with non-emp ty neighborho o ds are the ones in A ϕ . 14 Choice of parameters for BSG-Test : W e shall choose the paramete rs for the ab o v e test as follo ws. Recall that ρ = ε 16 / 4. W e tak e ρ 1 = 21 ρ 3 / 20 and ρ 2 = 19 ρ 2 / 20. Giv en an error parameter δ , w e tak e r and s to b e poly (1 /ρ, log (1 /δ )), so that with probabilit y at least 1 − δ , t h e error in the last tw o estimates is at most ρ 3 / 100. Also, b y using p oly(1 /ρ, log (1 /δ )) samples in eac h call to Edge-Tes t , w e can assume that th e er r or in all estimates used b y Edge- Test is at most ρ 3 / 100. T o c ho ose γ 1 , γ 2 , γ 3 , w e divid e the inte rv al [ ε 16 / 180 , ε 16 / 18] in to 4 /ρ 2 consecutiv e sub -in terv als of size ρ 3 / 20 eac h. W e then randomly choose a sub-in terv al and c ho ose positive parameters γ , µ so that γ − µ and γ + µ are endp oin ts of this in terv al. W e set γ 1 = γ 3 = γ + µ/ 2 and γ 2 = γ − µ/ 2. T o analyze BSG-Test , we “sandwic h” the elemen ts on whic h it answ ers 1 b et wee n a large set and a set w ith small d oubling. Lemma 4.10 L et δ > 0 and p ar ameters ρ 1 , ρ 2 , r , s b e chosen as ab ove. Then for every u = ( x, ϕ ( x )) and eve ry choic e of γ 1 , γ 2 , γ 3 as ab ove, ther e exist two sets A (1) ϕ ( u ) ⊆ A (2) ϕ ( u ) , such that the output of BSG-T est satisﬁes the fol lowing with pr ob ability at le ast 1 − δ . • BSG-Test ( u, v , γ 1 , γ 2 , γ 3 , ρ 1 , ρ 2 ) = 1 = ⇒ v ∈ A (2) ϕ ( u ) . • BSG-Test ( u, v , γ 1 , γ 2 , γ 3 , ρ 1 , ρ 2 ) = 0 = ⇒ v / ∈ A (1) ϕ ( u ) . Mor e over, with pr ob ability ρ 3 / 24 over the choic e of u and γ 1 , γ 2 , γ 3 , we have | A (1) ϕ ( u ) | ≥ ( ρ/ 6) · N and | A (2) ϕ ( u ) + A (2) ϕ ( u ) | ≤ (2 /ρ ) 8 · N . Pro of: T o deal with the app ro ximate nature of Edge-Test , we deﬁne the follo wing sets: N γ ( u ) def = { v : ( u, v ) ∈ E γ } T ( u, γ 1 , γ 2 , γ 3 , ρ 1 , ρ 2 ) def =  v ∈ N γ 1 ( u ) : P v 1  v 1 ∈ N γ 2 ( u ) & P v 2 [ v 2 ∈ N γ 3 ( v ) ∩ N γ 3 ( v 1 )] ≤ ρ 1  ≤ ρ 2  Going through the d eﬁnitions and recall ing that E γ ⊆ E γ − γ ′ for γ ′ > 0, it can b e c hec ked that the s ets T ( u, γ 1 , γ 2 , γ 3 , ρ 1 , ρ 2 ) are m onotone in th e v arious parameters. In particular, for γ ′ 1 , γ ′ 2 , γ ′ 3 , ρ ′ 1 , ρ ′ 2 > 0 T ( u, γ 1 , γ 2 , γ 3 , ρ 1 , ρ 2 ) ⊆ T ( u, γ 1 − γ ′ 1 , γ 2 + γ ′ 2 , γ 3 − γ ′ 3 , ρ 1 − ρ ′ 1 , ρ 2 + ρ ′ 2 ) . Recall that w e h a v e γ 1 = γ 3 = γ + µ / 2 and γ 2 = γ − µ / 2, where [ γ − µ , γ + µ ] is a sub-inte r v al of [ ε 16 / 180 , ε 16 / 18] of length ρ 3 / 20. W e deﬁne the sets A (1) ϕ ( u ) and A (2) ϕ ( u ) as b elow. A (1) ϕ ( u ) def = T ( u, γ + µ, γ − µ, γ + µ, 11 ρ 3 / 10 , 9 ρ 2 / 10) A (2) ϕ ( u ) def = T ( u, γ , γ , γ , ρ 3 , ρ 2 ) By th e m onotonicit y pr op erty noted ab o v e, we ha v e that A (1) ϕ ( u ) ⊆ A (2) ϕ ( u ). Also, b y the c hoice of parameters r , s and th e n u m b er of samples in Edge -Test , w e kno w that with probabilit y 1 − δ , the error in all estimates used in BSG-Test is at most ρ 3 / 100. Hence, we get that with probabilit y at least 1 − δ , if BSG-Tes t answ ers 1, then the input is in A (2) ϕ and if BSG -Test answ ers 0, then it is not in A (1) ϕ . It remains to pro v e the b oun d s on the size and doub ling of these sets. 15 By our c h oice of parameters, A (2) ϕ ( u ) is the same set as the one deﬁned in Sudako v et al. [SSV05]. They sh o w that if u is such that | A (2) ϕ ( u ) | ≥ 3 · ( ρ/ 2) 2 N , then | A (2) ϕ ( u ) + A (2) ϕ ( u ) | ≤ (2 / ρ ) 8 · N (see Lemma 3.2 in [Vio07] for a simpliﬁed pr o of of the v ersion menti oned here). T o sh o w the low er b ound on the size of A (2) ϕ ( u ), we will sh o w that in fact with probabilit y at least ρ 3 / 24 o v er the c hoice of u and γ 1 , γ 2 , γ 3 , we will ha ve | A (1) ϕ ( u ) | ≥ ( ρ/ 6) · N . Since A (1) ϕ ( u ) ⊆ A (2) ϕ ( u ), this s u ﬃces for the pro of. W e consider a slight mo diﬁcation of the argument of [SSV05], sh owing an u pp er bou n d on th e exp ected size of th e s et S ′ ( u ) deﬁned as S ′ ( u ) def = N γ + µ ( u ) \ T ( u, γ + µ, γ − µ, γ + µ, 11 ρ 3 / 10 , 9 ρ 2 / 10) =  v ∈ N γ + µ ( u ) : P v 1  v 1 ∈ N γ − µ ( u ) & P v 2 [ v 2 ∈ N γ + µ ( v ) ∩ N γ + µ ( v 1 )] ≤ 11 ρ 3 / 10  ≥ 9 ρ 2 / 10  . W e kno w from Lemma 4.6 that since γ + µ ≤ ε 16 / 18, th e quan tit y E u [ | N γ + µ ( u ) | ], whic h is the a v erage degree of the grap h , is a t lea st ρN (assuming that we are working with a go o d function ϕ ). Combining this w ith an upp er b ound on E u [ | S ′ ( u ) | ] will give the required lo we r b ound on the size of A (1) ϕ ( u ) = T ( u, γ + µ, γ − µ, γ + µ, 11 ρ 3 / 10 , 9 ρ 2 / 10). W e call a p air ( v, v 1 ) b ad if | N γ + µ ( v ) ∩ N γ + µ ( v ) | ≤ 11 ρ 3 N/ 10. W e need the follo w ing b ound. Claim 4.11 Ther e exists a choic e for the sub-interval [ γ − µ, γ + µ ] of length ρ 3 / 20 in [ ε 16 / 180 , ε 16 / 18] such that E u [# { b ad p airs ( v , v 1 ) : v ∈ N γ + µ ( u ) & v 1 ∈ N γ − µ ( u ) } ] ≤ 3 ρ 3 N 2 / 5 W e ﬁrst pro v e Lemma 4.10 assuming the claim. F r om the deﬁnition of S ′ ( u ), # { bad pairs ( v , v 1 ) : v ∈ N γ + µ ( u ) & v 1 ∈ N γ − µ ( u ) } ≥ | S ′ ( u ) | · (9 ρ 2 N/ 10) . Claim 4.11 giv es E u [ | S ′ ( u ) | ] ≤ (3 ρ 3 N 2 / 5) / (9 ρ 2 N/ 10) = (2 ρ/ 3) N , for at least one choice of the in terv al [ γ − µ, γ + µ ]. Since there are 4 /ρ 2 c hoices for the sub -interv al, this h app ens with pr obabilit y at least ρ 2 / 4. F or this c hoice of γ and µ (and hence of γ 1 , γ 2 , γ 3 ), w e also hav e E u [ | N γ + µ ( u ) | ] ≥ ρN . S ince S ′ ( u ) = N γ + µ ( u ) \ A (1) ϕ , we get that E u h | A (1) ϕ | i ≥ ρN − (2 ρ/ 3) N = ( ρ/ 3) N . Hence, with probab ility at least ρ/ 6 o ver the c hoice of u , | A (1) ϕ | ≥ ( ρ/ 6) N . Thus, w e obtain the desired outcome with probabilit y at least ρ 3 / 24 o v er the c hoice of u and γ 1 , γ 2 , γ 3 . Pro of of Claim 4.11: W e b egin b y obs erving that the exp ected num b er of bad pairs ( v, v 1 ) suc h that v ∈ N γ + µ ( u ) & v 1 ∈ N γ − µ ( u ) is equal to E u [# { bad pairs ( v, v 1 ) : v ∈ N γ + µ ( u ) & v 1 ∈ N γ + µ ( u ) } ] + E u [# { bad pairs ( v , v 1 ) : v ∈ N γ + µ ( u ) & v 1 ∈ N γ − µ ( u ) \ N γ + µ ( u ) } ] . Note that for ea ch of the  N 2  c hoices f or v , v 1 , if they form a bad pair, then eac h u is in N γ + µ ( v ) ∩ N γ + µ ( v 1 ) with pr obabilit y at most 11 ρ 3 / 10. Hence, the ﬁrst term is at most (11 ρ 3 / 20) N 2 . Also, the second term is at most N · E u [ | N γ − µ ( u ) \ N γ + µ ( u ) | ] = N ·  E u [ | N γ − µ ( u ) | ] − E u [ | N γ + µ ( u ) | ]  W e kn o w that E u [ | N γ ( u ) | ] is monotonically decreasing in γ . S ince it is at most N for γ = ε 16 / 180, there is at least one in terv al of size ρ 3 / 20 in [ ε 16 / 180 , ε 16 / 18], where the c hange is at most ρ 3 N/ 20. T aking γ + µ and γ − µ to b e the endp oin ts of th is in terv al ﬁn ishes the pr o of. 16 4.3 Obtaining a linea r choi ce function Using the subset giv en by the Balo g-Szemer ´ edi-Go wers theorem, one can use the somewhat linear c hoice function ϕ to ﬁn d an line ar tr ansformation x 7→ T x wh ich also selects large F our ier co eﬃ- cien ts in d er iv ativ es. In particular, it satisﬁes E x h ˆ f x 2 ( T x ) i ≥ η for some η = η ( ε ). This map T can then b e used to ﬁn d an appr op r iate quadratic phase. In this su bsection, w e giv e an algorithm for ﬁnding suc h a tr an s formation, using the pr o cedure BSG-Test deve lop ed ab o ve . In the lemma b elo w, w e assu me as b efore that ϕ is a go o d fu nction satisfying the guarante e in Lemma 4.6. W e also assu me that w e h a v e c hosen a go o d vertex u and parameters γ 1 , γ 2 , γ 3 satisfying the guaran tee in Lemma 4.10. Lemma 4.12 L et ϕ b e as ab ove and δ > 0 . Then ther e exists an η = exp( − 1 /ε C ) and an algorithm which makes O ( n 2 log n · p oly(1 /η , log (1 /δ ))) c al ls to BSG-Test and uses additional running time O ( n 3 ) to output a line ar map T or the symb ol ⊥ . If BSG-Test is deﬁne d using a go o d u and p ar ameters γ 1 , γ 2 , γ 3 as ab ove, then with pr ob ability at le ast 1 − δ the algorithm outp uts a m ap T satisfying E x h ˆ f x 2 ( T x ) i ≥ η . Pro of: Let t = 4 n 2 + log (10 /δ ). W e proceed by ﬁrst sampling K = 100 t/ρ elements ( x, ϕ ( x )) and run ning BS G-Test ( u, · ) on eac h of them with parameters as in Lemma 4.10 and δ ′ = δ / (5 K ). W e retain only the p oin ts ( x, ϕ ( x )) on whic h BS G-Test outputs 1. Since δ ′ = δ / (5 K ), BSG-Test do es not satisfy the g uarantee of Lemma 4 .10 on some qu er y with probabilit y at most δ / 5. W e assume this d o es not h app en for any of the p oints we sampled. If BSG-Test outp u ts 1 on few er than t of the q u eries, w e stop and outpu t ⊥ . The follo w ing claim sho ws that th e pr obabilit y of this happ enin g is at most δ/ 5. In fac t, the claim sho ws that w ith probabilit y 1 − δ / 5 there must b e at le ast t s amples fr om A (1) ϕ itself, on wh ic h we assumed t h at BSG-Test outputs 1. Claim 4.13 Wi th pr ob ability at le ast 1 − δ / 5 , the sample d p oints c ontain at le ast t samples fr om A (1) ϕ . Pro of: Since | A (1) ϕ | ≥ ρN / 6, the exp ected n umb er of samples from A (1) ϕ is at least ρK/ 6. By a Ho eﬀding b ound , the prob ab ility that this num b er is less than t is at most exp( − Ω( ρK )) ≤ δ / 5 if ρK = Ω (log(1 /δ )). Note that conditioned on being in A (1) ϕ , the sampled p oints are in fact uniformly distr ibuted in A (1) ϕ . W e sh o w that then they m ust span a subspace of large d imension, and that their s p an m ust co v er at least half of A (1) ϕ . Claim 4.14 L et z 1 , . . . , z t ∈ A (1) ϕ b e uniformly sample d p oints. Then for t ≥ 4 n 2 + O (log (1 /δ )) it is true with pr ob ability 1 − δ / 5 that • | < z 1 , . . . , z t > ∩ A (1) ϕ | ≥ (1 / 2) | A (1) ϕ | • dim( < z 1 , . . . , z t > ) ≥ n − log(12 /ρ ) . 17 Pro of: F or the ﬁ rst part, w e consider the sp an < z 1 , . . . , z t > , which is a s ubspace of F n 2 . Th e probabilit y that it has s m all in tersection with A (1) ϕ is X | S ∩ A (1) ϕ |≤| A (1) ϕ | / 2 P [ z 1 , . . . , z t ∈ S ] · P [ < z 1 , . . . , z t > = S | z 1 , . . . , z t ∈ S ] , where the sum is tak en o v er all subspaces S of F n 2 . Since | S ∩ A (1) ϕ | ≤ | A (1) ϕ | / 2, we ha v e that P [ z 1 , . . . , z t ∈ S ] ≤ (1 / 2) t . Th us, the r equired probabilit y b ound ed abov e b y X | S ∩ A (1) ϕ |≤| A (1) ϕ | / 2 (1 / 2) t · 1 ≤ 2 − t O (2 4 n 2 ) . The last b oun d uses the fact that the n umb er of su bspaces of F 2 n 2 is O (2 4 n 2 ). Th us, for t = 4 n 2 + log(10 /δ ), the pr ob ab ility is at most δ/ 10. W e no w b oun d th e probabilit y that the sampled p oint s z 1 , . . . , z t span a sub space of d imension at most n − k . The probabilit y that a random of A (1) ϕ lies in a sp e ciﬁc subspace of dimension n − k is at most (2 − k / ( ρ/ 6)) . Hence, th e probabilit y that all t p oints lie in an y subsp ace of dimension n − k is b ounded ab o ve by  2 − k ρ/ 6  t · # { subspaces of dim n − k } ≤  2 − k ρ/ 6  t · 2 n ( n − k ) . F or t ≥ n 2 + O (log (1 /δ )) and k = log (12 /ρ ), this p robabilit y is at most δ / 10. Hence the dimens ion of the sp an of the sampled v ectors is at least n − log(12 /ρ ) with high p robabilit y . Next, w e upp er b ound the dimension of the span of the retained p oints (on whic h B SG-Test answ ered 1). By the assumed correctness of BSG -Test , we get that all the p oints m u s t lie inside A (2) ϕ . App lying th e F r eiman-Ruzsa Theorem (Th eorem 2.4), it follo ws that | < A (2) ϕ > | ≤ exp(1 /ρ C ) N . The ab o ve implies that all the p oin ts are insid e a space of dimension at most n + log (1 /ν ), wh ere w e ha ve written ν = exp( − 1 /ρ C ). F rom here, w e can pro ceed in a similar fashion to [Sam07]. Let V denote the s p an of the retained p oints and let v 1 , . . . , v r b e a basis for V . W e can add v ectors to complete it to v 1 , . . . , v s so that the pro jection onto the ﬁrst n coordin ates has full rank. Let V ′ = < v 1 , . . . , v s > . W e can also assume, by a c hange of basis, that for i ≤ n we ha ve th e co ordinate v ectors v i = ( e i , u i ). This can all b e implemen ted by p erforming Gaussian eliminatio n , whic h tak es time O ( n 3 ). Consider the 2 n × s matrix with v 1 , . . . , v s as columns. By the p revious discussion, this matrix is of the form P =  I 0 T U  , where I is the n × n identit y matrix, and T and U are n × n and n × ( s − n ) matrices, resp ectiv ely . By Claim 4.14, w e kno w that v ′ con tains | A (1) ϕ | / 2 ≥ ( ρ/ 12) N v ectors of the form ( x, ϕ ( x )) T . F or eac h suc h v ector, there exists a w ∈ F s 2 suc h that P · w = ( x, ϕ ( x )) T . Because of the f orm of P , w e m us t ha v e that w = ( x, z ) f or z ∈ F s − n 2 . Th u s , w e get that for eac h v ector ( x, ϕ ( x )), w e in fact ha ve ϕ ( x ) = T x + U z for some z ∈ F s − n 2 . 18 Therefore, for at least one z 0 ∈ F s − n 2 and y 0 = U z 0 w e ﬁnd that P x ∈ F n 2 [ ϕ ( x ) = T x + y 0 ] ≥ ( ρ/ 12) · 2 − ( s − n ) . W e next u pp er b ound s − n . Note that s ≤ r + k since by Claim 4.14, V had dimension at least n − k for k = log(12 /ρ ). Also, we kno w that r ≤ n + log(1 /ν ) b y the b ound on | < A (2) ϕ > | , implying that s ≤ n + log(12 /ρ ) + log (1 /ν ). W e conclude that 2 − ( s − n ) ≥ ( ρ/ 12) ν . Moreo v er, for eac h element of the form ( x, ϕ ( x )) ∈ A (1) ϕ , w e kno w that | ˆ f x ( ϕ ( x )) | ≥ γ ≥ ε 16 / 180. This implies th at E x ∈ F n 2 h ˆ f x 2 ( T x + y 0 ) i ≥ γ 2 · ( ρ/ 12) · ( ρν / 12) . Samoro dnitsky sho ws that w e can in fact take y 0 to b e 0. In fact, h e sho ws the f ollo win g general claim. Claim 4.15 (Consequence of Lemma 6.10 [Sam07]) F or any matrix T and y ∈ F n 2 , E x ∈ F n 2 h ˆ f x 2 ( T x + y ) i ≤ E x ∈ F n 2 h ˆ f x 2 ( T x ) i . Th u s, w e s im p ly output th e matrix T constructed as ab o v e. F or η = γ 2 ρ 2 ν / 144, it satisﬁes E x ∈ F n 2 h ˆ f x 2 ( T x ) i ≥ η . Finally , w e calculate the p robabilit y that th e algorithm outputs ⊥ or outp uts a T n ot s atisfying this guaran tee. This can happ en only when th e guaran tee on B SG-Test is not satisﬁed for o n e o f the sampled points, or when the guaran tees in Claims 4 .13 and 4.14 are n ot satisﬁed. Sin ce eac h of these h app en with probabilit y at most δ / 5, the p robabilit y of error is at most 3 δ/ 5 < δ . 4.4 Finding a qu adratic phase function Once we ha ve identiﬁed the linear map T ab o v e, the remaining argument is identica l to the one in [Sam07]. Equipp ed with T , o ne can ﬁn d a symmetric matrix B w ith zero diagonal that satisﬁes a sligh tly w eak er guaran tee. This step is usually referred to as the symmetry ar gument , and we shall encounter a m o diﬁcation of it in Section 5 . Th e only algo rithm ic steps us ed in the pro cess are Gaussian elimination and ﬁnding a basis for a sub space, wh ic h can b oth b e done in time O ( n 3 ). Lemma 4.16 (Pro of o f Theorem 2.3 [Sam07 ]) L et T b e as ab ove. Then in time O ( n 3 ) one c an ﬁnd a symmetric matrix B with zer o diagona l suc h that E x ∈ F n 2 h ˆ f x 2 ( B x ) i ≥ η 2 . No w that w e hav e correlati on of the der iv ativ e f x of th e function with a truly linear map, it remains to “in tegrate” this r elationship to obtain that f itself correlates w ith a quadratic map. F ollo wing Green and T ao, w e shall henceforth refer to this part of the argument as the inte gr ation step . Ha ving obtained B ab o ve, w e can ﬁnd a matrix M suc h that M + M T = B . W e tak e the quadratic part of the phase function to b e h ( x ) = ( − 1) h x,M x i . The follo wing claim helps establish the linear part. Lemma 4.17 (Corollary 6.4 [Sam07 ]) L et B and h b e as ab ove. Then ther e exists α ∈ F n 2 such that | c f h ( α ) | ≥ η 2 . 19 An appropr iate α can b e f ound using the algorithm Linear -Decompo sition with parameter γ ′ = η 2 (b y pic king an y ele ment from the list it outputs). W e tak e q ( x ) = h x, M x i + h α, x i + c w here ( − 1) c is the sign of the coeﬃcien t for ( − 1) h α,x i in the li n ear decomp osition. The runnin g time of this step is O ( n 3 log n · p oly(1 /η , log (1 /δ ))), where δ is the probabilit y of error we wan t to allo w for this in vocation of Linear-D ecomposi tion . Note that of all the steps in v olve d in ﬁ nding a qu adratic phase, ﬁnding th e line ar part of the phase is the only step for whic h r unnin g time dep ends exp onen tially on ε (since η = exp( − 1 /ε Ω(1) )). The runn in g time of all other steps dep ends p olynomially on 1 / ε . 4.5 Putting things together W e are no w ready to ﬁn ish th e pr o of of Theorem 4.1. Pro of of Theorem 4.1: F or the pro cedure Find-Qua dratic the fu nction ϕ ( x ) will b e sampled using Lemma 4.6 as required . W e start with a random u = ( x, ϕ ( x )) and a random c hoice for the parameters γ 1 , γ 2 , γ 3 as describ ed in the analysis of BSG -Test . W e run the algorithm in Lemma 4.12 using B SG-Test with the ab ov e parameters and with error p arameter 1 / 2. If the algorithm outpu ts a quadratic form q ( x ), we estimate |h f , ( − 1) q i| using O ((1 /η 4 ) · log 2 ( ρ/δ )) samples. If th e estimate is less than η 2 / 2, or if the algorithm stopp ed with output ⊥ we d iscard q and rep eat the en tire p ro cess. F or a M to b e c h osen later, if w e do not ﬁn d a qu adratic ph ase in M attempts, we stop and output ⊥ . With probabilit y ρ/ 2, all samples of ϕ ( x ) (sampled with error 1 /n 5 ) corresp ond to a goo d fu nction ϕ . Conditioned on this, we h a v e a goo d choice of u and γ 1 , γ 2 , γ 3 for BSG-T est with probabilit y ρ 3 / 24. Conditioned on b oth the ab o v e, the alg orithm in Lemma 4.12 ﬁn ds a goo d transformation with p robabilit y 1 / 2. Th us, for M = O ((1 /ρ 4 ) · log(1 /δ )), th e algo rith m stops in M attempts with probabilit y at least 1 − δ / 2. By choic e of th e num b er o f samples ab ov e, the probabilit y that w e estimate |h f , ( − 1) q i| incorrectly at an y step is at most δ / 2 M . Thus, with pr obabilit y at least 1 − δ , w e output a goo d quadratic phase. One call to the alg orithm in Lemma 4.12 requires O ( n 2 ) calls to B SG-Test , which in turn requires p oly(1 /ε ) calls to Linear- Decompos ition , eac h taking time O ( n 2 log n ). Th is dominates the runn in g time of the algorithm, which is O ( n 4 log n · p oly(1 /ε, 1 /η , log (1 /δ ))). 5 A reﬁnemen t of the in v erse theorem In this section we shall wo rk with a n umb er of reﬁn emen ts of the inv erse theorem as stated in Theorem 2.6. F or the pur p oses of the preliminary discussion w e shall th ink of p b eing any prime, and later sp ecialize to th e case p = 2. It w as observed (b ut not exploited) b y Green and T ao [GT08] that a sligh tly stronger form of the in ve rs e theorem h olds. If V is a subsp ace of F n p and y ∈ F n p , then one can d eﬁ ne a seminorm k . k u 3 ( y + V ) on functions f r om F n p to C by setting k f k u 3 ( y + V ) = sup q | E x ∈ y + V f ( x ) ω − q ( x ) | , where the supr em um is tak en o v er all quadratic forms q on y + V and ω denotes a p th root of un ity . This semi-norm measures the correlatio n ov er a co set of th e sub space V . W e shall b e interested in 20 the co -dimen s ion of the su bspace, whic h we shall denote by co d V . With this notation, the inv erse theorem in [GT08] can b e stated as f ollo ws. Theorem 5.1 (Lo cal I n v erse T he orem for U 3 [GT08]) L et p > 2 , and let f : F n p → C b e a function such that k f k ∞ ≤ 1 and k f k U 3 ≥ ε . Then ther e exists a subsp ac e V of F n p such that co d V ≤ ε − C and E y ∈ V ∗ k f k u 3 ( y + V ) ≥ ε C . Here w e ha ve denoted th e set of coset repr esen tativ es of V by V ∗ , s o that V ⊕ V ∗ = F n 2 . Actuall y , the theorem as u sually stated in v olv es an a ve rages o v er the whole of F n p as opp osed to just V ∗ , but the result can b e obtained with this mod iﬁcation without diﬃcult y b y a verag ing o v er coset represent ativ es th roughout the pro of. One can d educe the usual inv erse theorem from this v ersion without to o m uch eﬀort: by an a v er- aging argumen t, there must exist y such th at f correlates w ell on y + V with some quadratic phase function ω q ; this function ca n b e extended to a function on the whole of F n p in man y diﬀeren t w a ys, and a fur ther a v eraging argumen t yields the u sual b oun ds. Ho wev er, extending the qu adratic phase results in an exp onential loss in correlation. (See, f or example, Prop osition 3.2 in [GT08].) It tur n s out that, as Green and T ao remark, an eve n more precise theorem holds. The result as stated tells us that for eac h y we can ﬁnd a lo cal quadratic phase fun ction ω q y deﬁned on y + V suc h that the av erage of | E x ∈ y + V f ( x ) ω q y ( x ) | is at least ε C . Ho wev er, it is actually p ossible to do this in suc h a wa y that the quadratic parts of the qu adratic phase functions q y are the same. More precisely , it can b e done in suc h a w a y that eac h q y ( x ) has the form q ( x − y ) + l y ( x − y ) for a single quadratic function q : V → F p (that is indep en den t of y ) and some F r eiman 2-homomorphisms l y : V → F p . This p ar al lel c orr elation w as hea vily exploited b y Go wers and the seco nd author [GW1 0a , GW10b] in a series of p ap ers on what they called the true c omplexity of a s ystem of linear equations, leading to r adically impro ve d b oun ds compared with the original approac h in [GW10c], whic h w as based on an ergodic-st yle decomp osition theorem due to Green and T ao [Gre07]. F or p = 2, the equiv alen t of Theorem 5.1 follo ws directly n either from Gr een an d T ao’s nor Samoro d- nitsky’s approac h b u t instead requires a merging of the t wo . The Green-T ao approac h is n ot d ir ectly applicable since the so-called symmetry ar gument in that pap er uses division by 2, while Samorod - nitsky’s approac h loses th e local information after an applicatio n of F reiman’s theorem. Section 5 is dedicated to showing how to obtain this lo c al c orr elation 3 in the case wh ere the c h aracteristic is equal to 2. W e shall therefore r estrict our atten tion to this case for th e remainder of the d iscu ssion, b earing in mind that it applies almost verbatim to general p . In order to b e able to refer to the parallel co rr elation prop erty more concisely , w e shall use the concept of quadr atic aver ages introdu ced in [GW10a]. As explained ab o ve , for eac h coset y + V , y ∈ V ∗ , w e can sp ecify a quad r atic phase q y ( x ) = q ( x − y ) + l y ( x − y ). W e extend the deﬁnition of q y to all y ∈ F n p b y setting them equal to q ˆ y where ˆ y ∈ V ∗ is suc h that y ∈ ˆ y + V . No w we can deﬁne a quadratic a verage via th e f orm ula Q ( x ) = E y ∈ x − V ( − 1) q y ( x ) . 3 The term “lo cal correlation” may b e slightly confusing. It is often u sed to refer to the fact that in Z / N Z , n o global quadratic correlation with a qu adratic phase can b e guaranteed. I ndeed, such a phase fun ction must b e restricted to a Bohr set, or the correlation assumed to only take place on a long arithmetic progressi on, as in Gow ers’s origi nal w ork. H ow ever, in F n p , th e setting w e are wo rkin g in here, th ere should be no ambiguit y . 21 Notice that the q y are th e same whenev er the y lie in the same coset of V . So in fact, since all the q y s occurr in g here are such that y ∈ x + V , they are all identic al. T h us the v alue of the quadratic a v erage only dep ends on the coset of V that x lies in. More precisely , we can write Q ( x ) = X y ∈ V ∗ 1 y + V ( x )( − 1) q y ( x ) . This tells u s that at most | V ∗ | man y linear phases are needed to sp ecify the quadratic a ve r age. Com binin g the Green-T ao approac h with Samoro dnitsky’s symmetry argumen t in c haracteristic 2, we shall obtain an algo rith m ic v ersion of the analogue of the Local In v erse Theorem (Theorem 5.1) for p = 2. In order to u se this result in our decomp osition algorithm Theorem 3.1, w e in fact state it as an algorithm for ﬁn ding a quadr atic aver age Q ( x ) = P y ∈ V ∗ 1 y + V ( x )( − 1) q y ( x ) , whic h has correlation p oly( ε ) with th e giv en fun ction. Using this, Theorem 3.1 will then yield a decomp osition in to p oly(1 /ε ) qu adratic av erages. F ollo wing [GW10c], w e shall call the codimen sion of V the c omplexity of the quadratic a verage . W e will ﬁnd qu adratic a verag es with complexit y p oly(1 /ε ). Note that while th is means that the description of a quadr atic av erage is still of size exp(1 /ε ), the d iﬀerent quadratic forms app earing in a qu adratic a verage only d iﬀer in the linear part. Theorem 5.2 Given ε, δ > 0 and n ∈ N , ther e exist K, C = O (1) and a r andomize d algorithm Find-Qua draticAv erage running in tim e O ( n 4 log 2 n · exp(1 /ε K ) · log (1 /δ )) , which, given or acle ac c ess to a function f : F n 2 → {− 1 , 1 } , either outputs a quadr atic aver age Q ( x ) of c omplexity O ( ε − C ) , or the symb ol ⊥ . The algorithm satisﬁes the fol lowing guar ante e : • If k f k U 3 ≥ ε , then with pr ob ability at le ast 1 − δ it ﬁnds a quadr atic aver age Q of c omplexity O ( ε − C ) such that h f , Q i ≥ ε C . • The pr ob ability that the algorithm outputs a Q which has h f , Q i ≤ ε C / 2 is at most δ . W e brieﬂy outline the k ey mod iﬁ cations in the p r o of that allo w us to obtain this resu lt. Recall that in the previous s ection we only ob tained correlation η = exp(1 /ε C ) b ecause we applied the F reiman- Ruzsa theorem to the set A (2) ϕ : w e w ere only able to assert that | < A (2) ϕ > | ≤ exp(1 /ε C ) | A (2) ϕ | . Because w e had correlation p oly( ε ) ov er A (2) ϕ , w e obtained correlatio n exp( − 1 /ε C ) with the linear function w e deﬁned on < A (2) ϕ > . They k ey diﬀerence in the n ew argumen t, w hic h b orro ws hea vily fr om Green and T ao [GT08], is that instead of lo oking for a subs pace c ontaining A (2) ϕ , whic h w e previously used to ﬁ nd a linear function, w e will lo ok for a subspace i nside 4 A (2) ϕ . Giv en the prop erties of A (2) ϕ , w e will b e able to ﬁnd such a subspace by an application of Bogolyub ov’s lemma (describ ed in more detail b elo w), with the prop ert y that the co-dimension of the subspace is p oly(1 /ε ). W e will also ﬁnd a quadr atic form suc h that r estricte d to inputs fr om this subsp ac e , it has correlation p oly(1 /ε ) w ith the fun ction f . W e shall then sho w ( Lemma 5.1 8 ) ho w to extend this quadratic form to all the co sets of the subspace, b y adding a diﬀer ent line ar fo rm for e ach c oset so that the correlation of the resulting quadratic a v erage is still p oly(1 /ε ). W e b egin b y dev eloping algorithmic v ersion of some of the new ingredien ts in the pro of. 22 5.1 An algorithmic v ersion of Bogolyubov’s lemma W e follo w Green a nd T ao in us in g a form of Bogolyub ov’s lemma, w hic h has b ecome a standard to ol in arithmetic com binatorics. Bogolyub o v’s lemma as it is usually s tated allo ws one to ﬁnd a large subspace inside the 4-fold sumset of an y giv en set of large size. W e brieﬂy remind the reader of the r elationship b et we en su m sets and con vo lutions, whic h is used in the pro of of the lemma. F or functions h 1 , h 2 : F n 2 → R , w e deﬁne their con v olution as h 1 ∗ h 2 ( x ) def = E y [ h 1 ( y ) h 2 ( x − y )] . The F ourier transform diagonalizes th e con vo lution op erator, that is, \ h 1 ∗ h 2 ( α ) = c h 1 ( α ) c h 2 ( α ) for an y t wo fun ctions h 1 , h 2 and any α ∈ F n 2 , which is easy to v erify fr om the deﬁnition. Also, if 1 A is the indicator function f or a set A ⊆ F n 2 , then 1 A ∗ 1 A ( x ) = E y [1 A ( y ) · 1 A ( x − y )] = |{ ( y 1 , y 2 ) : y 1 , y 2 ∈ A and y 1 + y 2 = x }| / 2 n . In particular, 1 A ∗ 1 A is supp orted only on A + A and giv es the n um b er of represen tations of x as the sum of t wo ele ments in A . In general, the k -fold conv olution is supp orted on the k -fold sumset. The pro of of Bogo lyub o v’s lemma constructs an explicit subspace by lo oking at th e large F ou r ier co eﬃcien ts (using the Goldreic h -Levin theorem) and sh o ws th at the 4-fold con vo lution is p ositiv e on this su bspace. Since w e will act u ally apply th is lemma not to a sub set bu t to the output of a randomized algo rithm , we state it for an arbitrary function h and its con v olution. W e will output a subspace V ⊆ F n 2 b y sp ecifying a basis for the sp ace V ⊥ def = { x : x T y = 0 ∀ y ∈ V } . Since ( V ⊥ ) ⊥ = V , this will also giv e us a w a y of c hec king if x ∈ V : w e sim p ly test if x T y = 0 f or all basis vec tors y of V ⊥ . Lemma 5.3 (Bogolyub o v’s Lemma) Ther e exists a r andomize d algorith m Bogo lyubov with p ar ameters ρ and δ which , given or acle ac c ess to a function h : F n 2 → { 0 , 1 } with E h ≥ ρ , outputs a subsp ac e V 6 F n 2 (by giving a b asis for V ⊥ ) of c o dimension at most O ( ρ − 3 ) suc h that with pr ob ability at le ast 1 − δ , we have h ∗ h ∗ h ∗ h ( x ) > ρ 4 / 2 for al l x ∈ V . The algorith m runs in time n 2 log n · p oly(1 /ρ, log (1 / ) . ) . Pro of: W e shall use the Goldreic h-Levin algorithm L inear-De composit ion for the fu nction h with p arameter γ = ρ 3 / 2 / 4 and error δ to pro duce a list K = { α 1 , . . . , α k } of length k = O ( γ − 2 ) = O ( ρ − 3 ). W e tak e V to b e the subspace { x ∈ F n 2 : h α, x i = 0 ∀ α ∈ K } and outpu t h K i . C learly co d ( V ) ≤ | K | . W e n ext consid er th e con vo lution h ∗ h ∗ h ∗ h ( x ) = X α | b h ( α ) | 4 ( − 1) h α,x i = X α ∈ K | b h ( α ) | 4 ( − 1) h α,x i + X α 6∈ K | b h ( α ) | 4 ( − 1) h α,x i . If x ∈ V , then X α ∈ K | b h ( α ) | 4 ( − 1) h α,x i + X α 6∈ K | b h ( α ) | 4 ( − 1) h α,x i ≥ | b h (0) | 4 − sup α / ∈ K | b h ( α ) | 2 · ρ The ﬁn al part of the guarante e in Theorem 4.4 states that the probabilit y of a F our ier co eﬃcient b eing larger than γ and not b eing on our list K is at most δ . W e conclude that with probabilit y at least 1 − δ , the expression h ∗ h ∗ h ∗ h ( x ) is b ounded b elo w, for all x ∈ V , b y ρ 4 − ρ · ρ 3 / 2 ≥ ρ 4 / 2 , and th us strictly p ositiv e. 23 W e will, in fact, n eed a fu rther t wist of the ab ov e lemma. The fun ction h to w hic h w ill app ly Lemma 5.3 will b e deﬁned by the outpu t of a randomized algorithm. Thus, h can be thought of as a rand om v ariable, where we c ho ose the v alue h ( x ) on eac h input x by ru nning the randomized algorithm. As in the case of BSG-Test , we will hav e the guarantee that there exist t w o sets A (1) ⊆ A (2) and δ ′ > 0 su c h that for e ach input x , w ith p r obabilit y 1 − δ ′ (o v er the c hoice of h ( x )) w e hav e 1 A (1) ( x ) ≤ h ( x ) ≤ 1 A (2) ( x ). W e w ill w an t to u s e this to conclude that for the entir e subsp ac e V giv en b y the algorithm Bog olyubov , V ⊆ 4 A (2) . T o argue th is, it will b e useful to consider the function h ′ deﬁned as h ′ def = min { 1 A (2) , max { h, 1 A (1) }} . By deﬁnition, we alw ays ha v e that 1 A (1) ( x ) ≤ h ′ ( x ) ≤ 1 A (2) ( x ). Also, if for eac h x , we ha v e with probabilit y 1 − δ ′ 1 A (1) ( x ) ≤ h ( x ) ≤ 1 A (2) ( x ), this means that for eac h x , P [ h ( x ) 6 = h ′ ( x )] ≤ δ ′ . The follo wing claim give s the desired conclusion for the subsp ace give n b y the algorithm Bogo lyubov . Claim 5.4 L et h b e a r andom function su c h that for δ ′ > 0 and for sets A (1) ⊆ A (2) ⊆ F n 2 , we have that for e v ery x with pr ob ability at le ast 1 − δ ′ , 1 A (1) ( x ) ≤ h ( x ) ≤ 1 A (2) ( x ) . Also, let E 1 A (1) ≥ ρ . L et h ′ = m in { 1 A (2) , max { h, 1 A (1) }} L et V b e the subsp ac e r eturne d by the algorithm Bogolyub ov when run with or acle ac c ess to h and err or p ar ameter δ . Then with pr ob ability at le ast 1 − δ − δ ′ · n 2 log n · p oly (1 /ρ, log (1 /δ )) , we have that for al l x ∈ V , 1 A (2) ∗ 1 A (2) ∗ 1 A (2) ∗ 1 A (2) ( x ) ≥ h ′ ∗ h ′ ∗ h ′ ∗ h ′ ( x ) > ρ 4 / 2 . In p articular, with ab ove pr ob ability, V ⊆ 4 A (2) . Pro of: C onsider the b eha vior of the algorithm B ogolyubov w hen r u n with oracle access to h ′ instead of h . Since it is alwa ys true that h ′ ≤ 1 A (2) and E [ h ′ ] ≥ E [1 A (1) ] ≥ ρ , the algorithm outputs, with p robabilit y 1 − δ , a s ubspace V such that for ev ery x ∈ V , 1 A (2) ∗ 1 A (2) ∗ 1 A (2) ∗ 1 A (2) ( x ) ≥ h ′ ∗ h ′ ∗ h ′ ∗ h ′ ( x ) > ρ 4 / 2. Thus, w ith prob ab ility 1 − δ , it outputs a subspace V such that V ⊆ 4 A (2) . Finally , w e observ e that the probabilit y that the algorithm outputs diﬀeren t subspaces when run with oracle access to h and h ′ is small. The probability of ha ving diﬀeren t outputs is at most the probabilit y that h and h ′ diﬀer on an y of inp uts queried by the algorithm Bogolyubov . Since it run s in time n 2 log n · p oly(1 /ρ, log (1 / ) . ), this p robabilit y is at most δ ′ · n 2 log n · p oly(1 /ρ, log (1 / ) . ). Thus, ev en when r u n with oracle access to h , with probability at least 1 − δ − δ ′ · n 2 log n · p oly(1 /ρ, log (1 / ) . ), the algorithm B ogolyubo v outputs a s u bspace V ⊆ 4 A (2) . Next we requ ire a version of Pl ¨ unnec ke’s inequalit y in order to d eal with th e size of iterated su msets. F or a p ro of we refer the in terested r eader to [TV06], or the recent sh ort and elegan t proof by P etridis [P et11]. Lemma 5.5 (Pl ¨ unnec ke’s Inequalit y) L et B ⊆ F n 2 b e su ch that | B + B | ≤ K | B | for some K > 1 . Then for any p ositive inte ger k , we have | k B | ≤ K k | B | . 5.2 Finding a goo d mo del set Again, as in Sectio n 4 we ma y assu me that ϕ is a goo d fu nction satisfying the guaran tee in L emma 4.6. Recall that A ϕ = { ( x, ϕ ( x )) : x ∈ A } , where A wa s deﬁned to b e A = { x : | b f x ( ϕ ( x )) | ≥ γ } . W e will u se the routine BSG-Test d escrib ed in Section 4. W e assume w e ha v e c hosen a go o d verte x u and parameters γ 1 , γ 2 , γ 3 satisfying the guaran tee in Lemma 4.10 for BSG-Tes t . W e will n eed to restrict the s ets A (1) ϕ and A (2) ϕ giv en by Lemma 4.10 a bit more b efore w e can apply Bogolyub o v’s lemma to ﬁnd an appropriate sub space. Because the subs pace sits inside the sumset 4 A (2) ϕ , an elemen t of the su bspace is of the form ( x 1 + x 2 + x 3 + x 4 , ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 )). 24 Ho w eve r, un lik e tup les of the form ( x, ϕ ( x )), the second h alf of the tu ple ( ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 )) ma y not uniquely dep end on the ﬁrst ( x 1 + x 2 + x 3 + x 4 ). Since w e will require this un iqueness prop erty from our sub space, w e restrict our sets to get new sets A ′ (1) ϕ ⊆ A ′ (2) ϕ . These restrictions will satisfy the follo wing prop erty: for all tuples x 1 , x 2 , x 3 , x 4 and x ′ 1 , x ′ 2 , x ′ 3 , x ′ 4 satisfying x 1 + x 2 + x 3 + x 4 = x ′ 1 + x ′ 2 + x ′ 3 + x ′ 4 , we also hav e ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) = ϕ ( x ′ 1 ) + ϕ ( x ′ 2 ) + ϕ ( x ′ 3 ) + ϕ ( x 4 ) ′ . In other w ords, ϕ is a F r eiman 4-homomorphism on the ﬁrst n co ordinates of A ′ (2) ϕ . W e will, in fact, need to ensure that it is a F reiman 8-homomorphism in order to obtain a truly linear map. W e sh all obtain these restrictions by in tersecting the original sets with a subspace, w hic h will b e deﬁned usin g a r an d om linear map Γ : F n 2 → F m 2 and a random ele ment c ∈ F m 2 (for m = O (log (1 /ε )) ). This step is often called ﬁnding a go o d mo del , and app ears (in non-algorithmic form) as Lemma 6.2 in [GT08]. W e shall apply the restriction Γ( ϕ ( x )) = c to the elements v = ( x, ϕ ( x )) on whic h BSG -Test outpu ts 1. Since w e assume we h a v e already c hosen go o d parameters u, ρ 1 , ρ 2 , γ 1 , γ 2 , γ 3 for the routine BSG-Test , we hide these paramete rs in the description of the pro cedur e b elo w. Model-Te st (v, Γ, c) - Let v = ( y , ϕ ( y )). - Answer 1 if BSG-Test r eturns 1 on v and Γ( ϕ ( y )) = c , and 0 otherwise. W e shall ﬁrst sho w that there exist g o o d c hoices of Γ and c for our purp oses. Let A (2) ϕ b e the set pro vided b y Lemma 4.1 0 for a go o d c hoice of parameters. Let B ⊆ F n 2 \ { 0 } b e the set of all t suc h that (0 , t ) ∈ 16 A (2) ϕ . Claim 5.6 L et θ ′ = ε 2448 / 2 487 . The set B has size at most θ ′− 1 . Pro of: W rite (0 , B ) for the set of all (0 , b ) , b ∈ B . Since A (2) ϕ is of th e form ( x, ϕ ( x )) for some function ϕ , we h a v e | A (2) ϕ + (0 , B ) | = | A (2) ϕ || B | , but at the same time A (2) ϕ + (0 , B ) ⊆ 17 A (2) ϕ . By Lemma 5 .5 w e ha v e | 17 A (2) ϕ | ≤ (3(2 /ρ ) 9 ) 17 | A (2) ϕ | ≤ (2 181 /ρ 153 ) | A (2) ϕ | since A (2) ϕ has small sumset, and therefore | B | ≤ 2 181 /ρ 153 = θ ′− 1 , s ince ρ = ε 16 / 4. Claim 5.7 L et m = 2 ⌈ log 2 θ ′− 1 ⌉ . Then with pr ob ability at le ast 1/2 a r andom line ar map Γ : F n 2 → F m 2 is non-zer o on al l of B . Pro of: Let Γ : F n 2 → F m 2 b e a r andomly c hosen linear transformation. Let E t b e the eve nt that Γ( t ) = 0. Clearly P ( E t ) ≤ 2 − m for eac h t ∈ B , and th us the probabilit y that Γ is non-zero on all of B is P ( ∩ t ( E C t )) = P (( ∪ t E t ) C ) = 1 − P ( ∪ t E t ) ≥ 1 − P t P ( E t ) ≥ 1 − | B | 2 − m ≥ 1 / 2 b y choice of m . So with probabilit y at least 1 / 2 we h a v e a map Γ that is non-zero on B . Claim 5.8 L et θ = θ ′ 2 ρ/ 12 , wher e θ ′ is the c onstant obtaine d in Claim 5.6, tha t is, we set θ = ε 4912 / (3 · 2 977 ) . Fix a map Γ as in Claim 5.7. Then with pr ob ability at le ast θ a r andomly chosen element c ∈ F m 2 is suc h that the set A ′ (1) ϕ def = { ( x, ϕ ( x )) ∈ A (1) ϕ : Γ( ϕ ( x )) = c } has size at le ast θ N . 25 Pro of: The exp ected size of this set is at least | A (1) ϕ | / 2 m ≥ ( ρN/ 6) / ( θ ′− 2 ) ≥ ( θ ′ 2 ρ/ 6) N , so with probabilit y θ we ca n get it to b e of size at least θ N . W e shall of co ur se also deﬁne A ′ (2) ϕ def = { ( x, ϕ ( x )) ∈ A (2) ϕ : Γ( ϕ ( x )) = c } , and since A (1) ϕ ⊆ A (2) ϕ , w e ha ve a similar con tainment for the new subsets, immediate ly giving a similar lo w er b ou n d on the size of A ′ (2) ϕ . W e summarize the ab o v e claims in the follo w in g reﬁnement of Lemma 4.1 0 . Lemma 5.9 L et the c al ls to BSG-Test in Model-Test b e with a go o d choic e of p ar ameters u, ρ 1 , ρ 2 , γ 1 , γ 2 , γ 3 and with e rr or p ar ameter δ > 0 . Then, ther e exist two sets A ′ (1) ϕ ⊆ A ′ (2) ϕ , the output of Model-T est on input v = ( y , ϕ ( y )) satisﬁes the fol lowing with pr ob ability 1 − δ . • Model-Te st ( v, Γ , c ) = 1 = ⇒ v ∈ A ′ (2) ϕ . • Model-Te st ( v, Γ , c ) = 0 = ⇒ v / ∈ A ′ (1) ϕ . Mor e over, with pr ob ability θ / 2 over the choic e of Γ and c , we have | A ′ (1) ϕ | ≥ θ N and ϕ is a F reiman 8-homomorphism on A (2) , wher e we denote the pr oje ction of A ′ (2) ϕ onto the ﬁrst n c o or dinates by A (2) . Pro of: If Mode l-Test outputs 1, th en v = ( y , ϕ ( y )) ∈ A (2) ϕ with probabilit y 1 − δ and Γ( ϕ ( y )) = c , so v ∈ A ′ (2) ϕ . Similarly , if Model-T est outputs 0 then either BSG-Tes t ga ve 0 or Γ( ϕ ( y )) 6 = c , s o in an y case v 6∈ A ′ (1) ϕ . By Claims 5.8 and 5.7, with probabilit y at least θ / 2 o ve r the choice of Γ and c , | A ′ (1) ϕ | ≥ θ N and Γ is n on-zero on all of B . It remains to v erify that ϕ is a F reiman 8-homomorphism on A (2) in this case. F or any (0 , t ) ∈ 16 A ′ (2) ϕ , we ha ve t 6 = 0 ⇒ t ∈ B b y d eﬁnition. Also Γ( t ) = 16 c = 0 by linearit y of Γ. Sin ce Γ is non-zero on all of B , w e must hav e t = 0. W e also ha ve 16 A ′ (2) ϕ = 8 A ′ (2) ϕ + 8 A ′ (2) ϕ , an d so if we tak e (0 , t ) = ( x 1 + · · · + x 8 + x ′ 1 + . . . x ′ 8 , ϕ ( x 1 ) + · · · + ϕ ( x 8 ) + ϕ ( x ′ 1 ) + . . . ϕ ( x ′ 8 )), w e ha v e that x 1 + · · · + x 8 + x ′ 1 + . . . x ′ 8 = 0 implies ϕ ( x 1 ) + · · · + ϕ ( x 8 ) + ϕ ( x ′ 1 ) + . . . ϕ ( x ′ 8 ) = 0, m aking ϕ a F reiman 8-homomorphism on A (2) . 5.3 Obtaining a linea r choi ce function on a subspace As b efore, we no w iden tify a linear transform (actually , an aﬃne transform ) that selects large F ourier co eﬃcien ts in d eriv ativ es. Ho wev er, as opp osed to Sectio n 4 where we d eﬁned a linear transf orm on the whole of F n 2 , here we will ju st d eﬁ ne it on a c oset a subsp ac e V suc h that cod ( V ) = p oly(1 /ε ). In particular, w e w ill pro v e the follo wing lo cal v ersion of Lemma 4.1 2 . Lemma 5.10 L et ϕ b e as ab ove and let the p ar ameters for BSG- Test and Mode l-Test b e so tha t they satisfy the guar ante es of lemmas 4.10 and 5.9. L et δ > 0 and ε b e as ab ove. Then ther e exists 26 an algorithm running in time O ( n 4 log 2 n · exp(1 /ε K ) · log 2 (1 /δ )) which outputs with pr ob ability at le ast 1 − δ a subsp ac e V of c o dimension at most ε − C as wel l as a line ar line ar map x 7→ T x and c 1 , c 2 ∈ F n 2 satisfying E x ∈ V + c 1 h b f x 2 ( T x + T c 1 + c 2 ) i ≥ ε C . Throughout the argument that follo ws, w e s hall assu me that we ha v e already c hosen goo d param- eters for BS G-Test and Model-T est so that the conclusions of Lemmas 4.10 and 5.9 hold. W e also assume we h av e acc ess to a goo d function ϕ as giv en by L emm a 4.6. T o ﬁnd the subspace V w e will apply Bogol yu b o v’s le mma to the set iden tiﬁed b y the pro cedu re Model-Te st . W e s hall lo ok at the second half of the tup les in this subspace (coord inates n + 1 to 2 n ) to ﬁ nd a linear c hoice function. Let h : F n 2 → { 0 , 1 } b e the (random) function deﬁned by h ( y ) = 1 if M odel-Tes t ( u, ( y , ϕ ( y )) , Γ , c ) = 1 and 0 ot herw ise. The error parameter δ ′ for Model-Test is tak en to b e δ /n 3 . W e shall apply the algorithm Bogolyub ov from Lemma 5.3 with queries to h and with error parameter δ 1 = δ / 20. Note that the function h is deﬁned on p oin ts in F n 2 . Let A (1) and A (2) denote pro jection o n the ﬁrst n co ordin ates of th e sets A ′ (1) ϕ and A ′ (2) ϕ giv en b y Lemma 5.9. Since the last n co ord inates are a function (namely ϕ ) of the ﬁ rst n coord in ates, we also ha v e | A ′ (1) ϕ | ≥ θ N , for θ a fu nction of ε as deﬁ ned in Claim 5.8. Also, with probab ility 1 − δ ′ for eac h input x , th e inequalit y 1 A (1) ( x ) ≤ h ( x ) ≤ 1 A (2) ( x ) holds. By Claim 5.4, we obtain a subspace V 0 of codimens ion θ − 3 suc h that w ith probabilit y at least 1 − δ 1 − δ ′ · n 2 log n · p oly(1 /θ , log (1 /δ 1 )) > 1 − δ / 10 , w e h a v e V 0 ⊆ 4 A (2) . Thus, eac h elemen t x ∈ V 0 can w e written as x 1 + x 2 + x 3 + x 4 for x 1 , x 2 , x 3 , x 4 ∈ A (2) . W e next sho w that the set Z 0 def =  ( x 1 + x 2 + x 3 + x 4 , ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ))     x 1 + x 2 + x 3 + x 4 ∈ V 0 , x 1 , x 2 , x 3 , x 4 ∈ A (2)  is also a s ubspace of F 2 n 2 . Ob serv e that the v alue of ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) is uniquely determined b y x 1 + x 2 + x 3 + x 4 . Claim 5.11 Ther e exists a line ar map ζ : V 0 → F n 2 satisfying for any x 1 , x 2 , x 3 , x 4 ∈ A (2) such that x 1 + x 2 + x 3 + x 4 ∈ V 0 , we have ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) = ζ ( x 1 + x 2 + x 3 + x 4 ) . Thus, the set Z 0 c an b e written as Z 0 = { ( x, ζ ( x )) : x ∈ V 0 } and is a sub sp ac e of F n 2 . Pro of: W e ﬁrst sh o w that the v alue of ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) is u niquely determined b y x 1 + x 2 + x 3 + x 4 . By Lemma 5.9, w e kno w that ϕ is a F reiman 8-homomorphism on A (2) and hence it is also a F reiman 4-homomorphism. In particular, if for x 1 , x 2 , x 3 , x 4 ∈ A (2) and x ′ 1 , x ′ 2 , x ′ 3 , x ′ 4 ∈ A (2) , w e ha ve that x 1 + x 2 + x 3 + x 4 = x ′ 1 + x ′ 2 + x ′ 3 + x ′ 4 , then it also h olds that ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) = ϕ ( x ′ 1 ) + ϕ ( x ′ 2 ) + ϕ ( x ′ 3 ) + ϕ ( x ′ 4 ). T h us , w e can write th e set Z 0 as { ( x, ζ ( x )) : x ∈ V 0 } , where ζ if some function on V . W e next sh o w that ζ m ust b e a linear function. W e ﬁrst sh o w that ζ (0) = 0. Since 0 ∈ V 0 , we m ust hav e elemen ts x 1 , x 2 , x 3 , x 4 ∈ A (2) with the prop erty that x 1 + x 2 + x 3 + x 4 = 0, in other w ords, x 1 + x 2 = x 3 + x 4 . But since ϕ is also a F r eiman 2-homomorphism, we get that ϕ ( x 1 ) + ϕ ( x 2 ) = ϕ ( x 3 ) + ϕ ( x 4 ), whic h implies that ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) = ζ (0) = 0. Since ϕ is a F reiman 8-homomorphism on A (2) and V 0 ⊆ 4 A (2) , it foll ows that ζ is a F reiman 2- homomorphism on V 0 . Since V 0 is closed un der addition, f or x, y ∈ V 0 w e can wr ite x + y = 0 + ( x + y ) with all four sum mands in V 0 . Since ζ is 2-homomorphic, w e get that ζ ( x ) + ζ ( y ) = ζ (0) + ζ ( x + y ) = ζ ( x + y ). 27 W e would lik e to u se th e linear map ζ to obtain the c hoice function on a coset of the sp ace V 0 . Ho w eve r, the p roblem is that w e d o not know the function ζ . W e get around this obstacle b y generating rand om tuples ( x 1 + x 2 + x 3 + x 4 , ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 )) su c h that x 1 + x 2 + x 3 + x 4 and eac h x i ∈ A (2) . W e show that for suﬃcien tly man y samples, the sampled p oin ts span a large subspace V of V 0 . Since ϕ ( x 1 ) + ϕ ( x 2 ) + ϕ ( x 3 ) + ϕ ( x 4 ) = ζ ( x 1 + x 2 + x 3 + x 4 ) on V 0 , we will b e able to obtain the desired linear map on the subspace V . W e samp le a p oint as f ollo w s. F or the j th sample, we generate four pairs ( x j 1 , ϕ ( x j 1 )) , . . . , ( x j 4 , ϕ ( x j 4 )). W e accept the sample if all four pairs are ac cepted b y Mod el-Test an d if x j 1 + x j 2 + x j 3 + x j 4 ∈ V . If a sample is accepted, w e s tore the p oin t y j = x j 1 + x j 2 + x j 3 + x j 4 and ζ ( y j ) = ϕ ( x j 1 )+ ϕ ( x j 2 )+ ϕ ( x j 3 )+ ϕ ( x j 4 ). Note that mem b ership in V 0 can b e tested eﬃcien tly s in ce w e kno w th e basis for V ⊥ 0 . W e ﬁrst estimate the probability that a p oin t ( y , ζ ( y )) for y ∈ V 0 is accepted b y the ab ov e test. This also giv es a b ound on the num b er of samples to b e tried so that at least t = O ( n 2 ) samples are ac cepted. Claim 5.12 F or a y ∈ V 0 , the pr ob ability that a samp le is ac c epte d by the ab ove pr o c e dur e and the stor e d p air is e qual to ( y , ζ ( y )) is at le ast θ 4 / 4 N . Mor e over, for some suﬃciently lar ge c onstant C , the pr ob ability that out of C exp(1 /θ 3 ) · (1 /θ 4 ) · t · log(10 /δ ) sa mples fewer than t ar e ac c epte d is at most δ / 10 . Pro of: Since th e fun ction h ( x ) = 1 exactly when Model- Test acce pts ( x, ϕ ( x )), the probabilit y that a s ample ( x 1 , ϕ ( x 1 )) , . . . , ( x 4 , ϕ ( x 4 )) is accepted and that x 1 + x 2 + x 3 + x 4 = y , is equal to P " 4 ^ i =1 ( h ( x i ) = 1) ∧ ( x 1 + x 2 + x 3 + x 4 = y ) # = (1 / N ) · E h,x 1 + x 2 + x 3 + x 4 = y [ h ( x 1 ) h ( x 2 ) h ( x 3 ) h ( x 4 )] As in Clai m 5.4, we deﬁn e the function h ′ = m ax { 1 A (1) , min { h, 1 A (2) }} . As b efore, w e hav e that for eac h x , P [ h ( x ) 6 = h ′ ( x )] ≤ δ ′ , and that h ′ ∗ h ′ ∗ h ′ ∗ h ′ ( x ) > θ 4 / 2 for eac h x ∈ V 0 . W e can no w estimate the abov e exp ectatio n as E h,x 1 + x 2 + x 3 + x = y [ h ( x 1 ) h ( x 2 ) h ( x 3 ) h ( x 4 )] ≥ P h,x 1 + x 2 + x 3 + x 4 = y  ∧ 4 i =1 ( h ( x i ) = h ′ ( x i ))  · E h,x 1 ,x 2 ,x 3  h ′ ( x 1 ) h ′ ( x 2 ) h ′ ( x 3 ) h ′ ( y + x 1 + x 2 + x 3 )  ≥ (1 − 4 δ ′ ) · h ′ ∗ h ′ ∗ h ′ ∗ h ′ ( y ) ≥ (1 − 4 δ ′ ) · ( θ 4 / 2) ≥ θ 4 / 4 . The last in equ alit y exploited the fact that h ′ ∗ h ′ ∗ h ′ ∗ h ′ ( y ) ≥ θ 4 / 2 for y ∈ V 0 . The probabilit y that a samp le is accepted is equal to the p r obabilit y that one s elects a p air ( y , ζ ( y )) for some y ∈ V 0 . This is least ( | V 0 | / N ) · ( θ 4 / 2) = exp( − 1 /θ 3 ) · ( θ 4 / 2). The b ound on th e probabilit y of accepting few er than t samples is then giv en b y a Hoeﬀding b ound . Let ( y 1 , ζ ( y 1 )) , . . . , ( y t , ζ ( y t )) b e t stored p oin ts corresp onding to t samples accepted by th e ab o v e pro cedur e. The follo wing claim analog ous to Claim 4.14 shows that for t = O ( n 2 ), the pro jection on the ﬁ rst n co ordinates of these p oint s m us t span a large su bspace of V 0 . Claim 5.13 L et ( y 1 , ζ ( y 1 )) , . . . , ( y t , ζ ( y t )) b e t p oints stor e d ac c or ding to the ab ove pr o c e dur e. F or t = n 2 + log (10 /δ ) , the pr ob ability that cod ( < y 1 , . . . , y t > ) ≥ cod ( V 0 ) + log (4 /θ 4 ) is at most δ / 10 . 28 Pro of: Let k = co d ( V 0 ) + 4 log(4 /θ ) and let S b e any subs p ace of co d imension k . The p robabilit y that a sample ( x 1 , ϕ ( x 1 )) , . . . , ( x 4 , ϕ ( x 4 )) is accepted and has x 1 + x 2 + x 3 + x 4 = y for a sp eciﬁc y ∈ S is at most 1 / N . Thus, the probabilit y that an accepted samp le ( y j , ζ ( y j )) has y j ∈ S , conditioned on b eing accepted, is at most ( | S | / N ) / (( | V 0 | / N ) · ( θ 4 / 2)). Thus, the probab ility that all t stored p oin ts lie in any sub space of co-dimension k is at most  | S | / N ( | V 0 | / N ) · ( θ 4 / 2)  t · # { susp aces of co-dimension k } =  θ 4 / 4 θ 4 / 2  t · 2 n ( n − k ) ≤ 2 − t · 2 n 2 , whic h is at most δ / 10 for t = n 2 + log (10 /δ ). Let V = < y 1 , . . . , y t > . The ab o ve claim sho ws that with high pr obabilit y , the co d imension of V satisﬁes co d ( V ) = exp(1 /θ 3 ). F rom the w ay the samp les were generated, we also know ζ ( y 1 ) , . . . , ζ ( y t ). S ince ζ is a linear function b y Claim 5.11, we can extend it to a linear transform x 7→ T x suc h that ∀ x ∈ V , T x = ζ ( x ) (as in Section 4). W e now sho w that there is a coset of V on wh ic h T x iden tiﬁes large F our ier co eﬃcien ts of the deriv ative f x . W e deﬁne the set Z def = { ( x, T x ) : x ∈ V } . W e will ﬁnd a coset o f Z suc h that a signiﬁcan t fraction of p oin ts in this coset are of the form ( x, ϕ ( x )) ∈ A ′ (2) ϕ . Recall that a p oint ( x, ϕ ( x )) in A ′ (2) ϕ satisﬁes | ˆ f x ( ϕ ( x )) | ≥ γ = O ( ε 16 ). Th us, T x will b e a linear function sele cting large F ourier co eﬃcien ts f or a signiﬁcant fraction of p oints in this coset. The follo win g claim sho ws the existence of such a coset. Claim 5.14 The sets Z + A ′ (1) ϕ and Z + A ′ (2) ϕ b oth c onsist of at most (1 /θ ) · ( N / | Z | ) c osets of Z . Henc e , for some c ∈ A ′ (1) ϕ we have | ( Z + c ) ∩ A ′ (2) ϕ | ≥ | ( Z + c ) ∩ A ′ (1) ϕ | ≥ θ 2 · | Z | . Pro of: Since Z ⊆ 4 A ′ (2) ϕ and A ′ (1) ϕ ⊆ A ′ (2) ϕ , we ha v e that Z + A ′ (1) ϕ ⊆ Z + A ′ (2) ϕ ⊆ 5 A ′ (2) ϕ ⊆ 5 A (2) ϕ . The last inclusion follo ws from the fact that A ′ (2) ϕ w as obtained b y in tersecting A ′ (2) ϕ (giv en b y Lemma 4.1 0 ) with a s ubspace. W e kno w from L emm a 4.10 that | A (2) ϕ + A (2) ϕ | ≤ (2 /ρ ) 8 · N ≤ (2 /ρ ) 8 · (6 /ρ ) · | A (2) ϕ | . Lemma 5.5 (Pl ¨ unneck e’s inequalit y) then giv es that | 5 A (2) ϕ | ≤ (6 /ρ ) 45 · | A (2) ϕ | ≤ (1 /θ ) · | A (2) ϕ | ≤ (1 /θ ) · N . T hus, | Z + A ′ (2) ϕ | ≤ (1 /θ ) · N and it is the union of at most (1 /θ ) · ( N / | Z | ) cosets. Since A ′ (1) ϕ ⊆ Z + A ′ (1) ϕ , th ere m ust exist at least one coset Z + c for c ∈ A ′ (1) ϕ , s uc h that    ( Z + c ) ∩ A ′ (1) ϕ    ≥ | A ′ (1) ϕ | (1 /θ ) · ( N / | Z | ) ≥ θ 2 · | Z | , where the last inequ ality used the fact that | A ′ (1) ϕ | ≥ θ N , as guaran teed by Lemma 5.9. W e now show ho w to computationally iden tify this coset of Z . W e will simp ly sample a suﬃcien tly large n umb er of p oint s on which Model-Test answers 1. W e will th en divide the p oints into diﬀeren t cosets of Z and pic k the coset with the most num b er of elemen ts. The f ollo wing claim sho ws that this p ro cedure succeeds in ﬁ nding the d esired coset with high probabilit y . 29 Claim 5.15 L et s = C · ( N / | Z | ) · (log(1 /δ ) /θ 5 ) ≤ C · exp(1 /θ 3 ) · (lo g (1 /δ ) /θ 5 ) for a suﬃciently lar ge c onstant C . Ther e exists an algorithm which runs in time O ( n 3 · s 2 ) and ﬁnds, with pr ob ability at le ast 1 − δ / 5 , a p oint c ∈ A ′ (2) ϕ such that | ( Z + c ) ∩ A ′ (2) ϕ | ≥ ( θ 2 / 2) · | Z | . Pro of: W e sample s in dep end en t elemen ts of the form ( x, ϕ ( x )) and reject all the ones on w hic h Model-Te st outpu ts 0, where w e r u n Model-Te st with error paramete r δ ′ = δ / (10 s ). F or some r ≤ s , let ( x 1 , ϕ ( x 1 )) , . . . , ( x r , ϕ ( x r )) b e the accepted elemen ts. F or eac h i, j ≤ r , we test if ( x i , ϕ ( x i )) and ( x j , ϕ ( x j )) lie in the same coset of Z , by chec king if ( x i − x j , ϕ ( x i ) − ϕ ( x j )) ∈ Z . This tak es time O ( n 3 ) for eac h i, j as w e need to c heck if ( x i − x j , ϕ ( x i ) − ϕ ( x j )) can be expressed as a linear com bin ation of the basis ve ctors for Z , whic h r equires solving a s ystem of linear equations. Lying in the same coset is an equiv alence relat ion, whic h d ivides the p oint s ( x 1 , ϕ ( x 1 )) , . . . , ( x r , ϕ ( x r )) in to equiv alence classes. W e pick the class with the maxim u m n umb er of elemen ts. Since (0 , 0) ∈ Z , for any element ( x i , ϕ ( x i )) in this class, we can w rite t h e coset as Z + ( x i , ϕ ( x i )). W e th u s pick an a rb itrary elemen t of th e form ( x i , ϕ ( x i )) in the la rgest class and outp ut c = ( x i , ϕ ( x i )). The run ning time of the ab o v e algorithm is O ( s 2 · n 3 ). W e need to argue that with p robabilit y at least 1 − δ / 5, the coset Z + c with the m axim um num b er of samp les satisﬁes | ( Z + c ) ∩ A ′ (2) ϕ | ≥ ( θ 2 / 2) · | Z | . With probabilit y at least 1 − δ ′ · s = 1 − δ / 10, Model-T est answers 1 on all elemen ts in A ′ (1) ϕ and 0 on all elemen ts outside A ′ (2) ϕ . F or an y coset of the form Z + c , let N ( Z + c ) b e the num b er of samples th at land in the coset. Cond itioned on th e correctness of M odel-Tes t , we ha v e that for an y coset of the form Z + c , s · | ( Z + c ) ∩ A ′ (1) ϕ | N ≤ E [ N ( Z + c )] ≤ s · | ( Z + c ) ∩ A ′ (2) ϕ | N , whic h b y deﬁnition of s implies that C · log(1 /δ ) θ 5 · | ( Z + c ) ∩ A ′ (1) ϕ | | Z | ≤ E [ N ( Z + c )] ≤ C · log(1 /δ ) θ 5 · | ( Z + c ) ∩ A ′ (2) ϕ | | Z | . By a Ho eﬀd in g b ound , the probabilit y that N ( Z + c ) deviate s by an additiv e ( C / 4) · (log (1 /δ ) /θ 3 ) from the exp ectation is at most δ · exp( − C ′ (1 /θ 3 )) for any ﬁxed coset. Since the n u mb er of cosets is at most (1 /θ ) · exp (1 /θ 3 ) by Claim 5.1 4 , the pr obabilit y that on a ny coset N ( Z + c ) d eviates from the exp ectation by the ab o v e amount is at most δ · exp( − C ′ (1 /θ 3 )) · (1 /θ ) · exp(1 /θ 3 ) < δ/ 10 for an appr opriate v alue of C ′ . By Claim 5.14, w e kno w that there is a coset Z + c with | ( Z + c ) ∩ A ′ (1) ϕ | ≥ θ 2 | Z | and hence E [ N ( Z + c ) ] ≥ C · (log(1 /δ ) /θ 3 ). By the ab o ve deviation b ound, we sh ould h a v e that N ( Z + c ) ≥ (3 C / 4) · (log (1 /δ ) /θ 3 ) for this coset. Thus, the coset with the maxim um num b er of samples, s a y Z + c ′ , will certainly also satisfy N ( Z + c ′ ) ≥ (3 C / 4) · (log (1 /δ ) /θ 3 ). Again, b y the deviatio n b ound, it m ust be true that E [ N ( Z + c ′ )] ≥ ( C / 2) · (log (1 /δ ) /θ 3 ), and hence | ( Z + c ) ∩ A ′ (2) ϕ | ≥ θ 2 | Z | / 2. W e can no w com bine the previous argum en t to pro v e Lemma 5.10. Pro of of Lemma 5.10 : W e follo w the steps describ ed ab o v e to ﬁnd the sub space V 0 , and subsequently the subspace V together with the transformation T . This immediately yields the 30 subspace Z = { ( x, T x ) : x ∈ V } . C laim 5.15 ﬁnds c = ( c 1 , c 2 ) ∈ F n 2 suc h that a fraction of at least θ 2 / 2 of p oin ts ( y + c 1 , T y + c 2 ) in the coset Z + ( c 1 , c 2 ) are of the form ( x, ϕ ( x )) for ( x, ϕ ( x )) ∈ A ′ (2) ϕ , and so | ˆ f x 2 ( ϕ ( x )) | ≥ γ = O ( ε 16 ). Since ( y, T y + c 2 ) = ( x + c 1 , ϕ ( x )) for these p oints, w e ha ve T ( x + c 1 ) + c 2 = ϕ ( x ). This implies E x ∈ c 1 + V h b f x 2 ( T x + T c 1 + c 2 ) i ≥ ( θ 2 / 2) · γ 2 ≥ ε C . (3) The errors in the application of Bogol yu b o v’s lemma and in Claims 5.12, 5.13 and 5.15 add up to δ / 2 < δ . The ru nning time is d ominated by the C exp(1 /θ 3 ) · (1 /θ 4 ) · t · log (10 /δ ) calls to Model-T est in Claim 5.12 for t = O ( n 2 ). Since eac h ca ll to M odel-Test ta kes O ( n 2 log n · p oly(1 /ε ) · log ( δ /n 3 )) time, the to tal ru nning time is O ( n 4 log 2 n · exp( O (1 /θ 3 )) · log 2 (1 /δ )). F ourier analysis o v er a subspace T o b egin with we collect some basic facts ab out F ourier analysis o ver a sub space of F n 2 , whic h will b e required for the remaining part of the argum en t. Let f : F n 2 → R b e a function and let W ⊆ F n 2 b e a su bspace. W e deﬁn e the F ourier co eﬃcien ts of f w ith resp ect to the sub space as the correlation with a lin ear ph ase ov er the sub space. As in the case of F ourier analysis o v er F n 2 , it is ea sy to v erify that the functions { χ α } α ∈ W with χ α ( x ) def = ( − 1) h α,x i form an orthonormal b asis for functions from W to R with r esp ect to the inner pro du ct h f 1 , f 2 i W def = E x ∈ W [ f 1 ( x ) f 2 ( x )]. Thus the dual group ˆ W of these basis functions is isomorphic to W . As in the case of F n 2 , w e hav e Parsev al’s iden tit y sa ying that P α ∈ W h f , χ α i 2 W = E x ∈ W  f 2 ( x )  . It is easy to mo dify the p ro of o f the Goldreic h-Levin theorem so th at it can b e used to iden tify the linea r functions χ α for α ∈ W that ha v e large correlation w ith a Bo olean function f o v er a subspace W . W e omit the details. Theorem 5.16 (Goldreic h-Levin theorem for a subspace) L et γ , δ > 0 and W ⊆ F n 2 b e a given subsp ac e. Ther e is a r andomize d algorithm which, give n or acle ac c ess to a function f : F n 2 → {− 1 , 1 } , runs in time O ( n 2 log n · p oly(1 /γ , log (1 /δ ))) and outputs a list L = { α 1 , . . . , α k } with e ach α i ∈ W such that • k = O (1 /γ 2 ) . • P  ∃ α i ∈ L |h f , χ α i i W | ≤ γ / 2  ≤ δ . • P  ∃ α / ∈ L |h f , χ α i i W | ≥ γ  ≤ δ . 5.4 Finding a qu adratic phase on a su bspace In order to dedu ce the reﬁned in ve r s e theorem (Th eorem 5.1) for p = 2, we need to redo the symmetry argument and inte gration phase with this lo cal expression obtained in Lemm a 5.10. The mod iﬁcations to S amoro dnitsky’s approac h are relativ ely minor but w e giv e complete p r o ofs nonetheless. One signiﬁcant diﬀeren ce is that w e w ill need to tak e F ourier tr ansforms relativ e to subspaces. W e b egin b y obtaining a subspace W 6 V on wh ic h the m atrix T obtained in the previous step is symmetric, thereb y pro viding the “local” analogue of Lemma 4.16. 31 Lemma 5.17 (Symmetry Argument) Given a subsp ac e V and a line ar map T with the pr op erty that E x ∈ c 1 + V b f x 2 ( T x + z c ) ≥ ε C , we c an output a subsp ac e W 6 V of c o dimension at most log( ε − C ) inside V to gether with a symmetric matrix B on W with zer o diagona l such that E x ∈ c 1 + W b f x 2 ( B x + z c ) ≥ ε C in time O ( n 3 ) . Pro of: W e let g ( x ) = ( − 1) h x,T x + z c i and F ( x ) = b f x 2 ( T x + z c ), and b egin b y noting that by Lemma 6.1 1 in [Sam07], w e ha ve that g ( x ) = − 1 implies F ( x ) = 0. Therefore w e ha v e ε C ≤ E x ∈ c 1 + V b f x 2 ( T x + z c ) = E x ∈ c 1 + V g ( x ) F ( x ) = E x ∈ V g c 1 ( x ) F c 1 ( x ) , w e h a v e written h y ( x ) for the sh if t h ( x + y ). T aking the F our ier transform relativ e to the subspace V , w e obtain ε C ≤ ( X α ∈ b V c g c 1 ( α ) d F c 1 ( α )) 2 , and b y the Cauc h y-Sch warz inequalit y and P arsev al’s theorem this is b ound ed ab o v e by X α ∈ b V c g c 1 ( α ) 2 X α ∈ b V d F c 1 ( α ) 2 ≤ E x ∈ V g c 1 ∗ V g c 1 ( x ) . The latte r (local) con vol u tion can ea sily b e co mp uted: g c 1 ∗ V g c 1 ( x ) = E y ∈ V ( − 1) h x + y + c 1 ,T ( x + y )+ c 2 i ( − 1) h y + c 1 ,T y + c 2 i = g c 1 ( x )( − 1) h c 1 ,c 2 i E y ∈ V ( − 1) h ( T + T T ) x,y i . The ﬁnal exp ectation giv es the indicator f unction of the subs pace W ′ = { x ∈ V : h ( T + T T ) x, y i = 0 for all y ∈ V } , that is, W ′ is a linear sub space on which T is symmetric. Note that W ′ is the space of solutions of a linear system of equations, a basis of whic h can b e computed b y Gaussian elimination in time O ( n 3 ). W e denote the m ap that tak es x to T x for x ∈ W ′ b y B . W e hav e just sho wn that | E x ∈ V 1 W ′ ( x ) g c 1 ( x ) | ≥ ε C , and in particular since g is b oun ded, w e quic kly observe that W ′ has densit y at least ε C inside V . Th is means the codimen s ion can hav e gone up by at most log ( ε − C ), wh ich is negligible in the grand sc heme of thin gs. It remains to ensure that B has zero diagonal. Again this can b e r ectiﬁed in a small num b er of steps. Denote th is diagonal by v ∈ F n 2 . Let W = W ′ ∩ < v + z c > ⊥ if h c 1 , c 2 i = 0, otherwise in tersect W ′ with the (unique) coset of < v + z c > ⊥ . S ince h x, B x i = h x, v i ov er F 2 , w e h a v e that h x + c 1 , v + z c i = h x, B x + z c i + h c 1 , c 2 i , and th us by Lemma 6.11 in [Sam07] if x + c 1 ∈ W ′ but / ∈ W , that is, x + c 1 / ∈ < v + z c > ⊥ , th en b f x 2 ( B x + z c ) = 0. Hence w e obtain 2 E x ∈ c 1 + W b f x 2 ( B x + z c ) = E x ∈ c 1 + W ′ b f x 2 ( B x + z c ) , whic h yields the desired conclusion. 32 Finally , we need to p erform t h e inte gration. Th e procedu re is v ery similar to Lemma 4.1 7 , b ut again w e ha v e to w ork relativ e to a subspace. Lemma 5.18 (Integration Step) L et f : F n 2 → [ − 1 , 1] . L et B b e a symmetric n × n matrix with zer o diagonal such that E x ∈ c 1 + W b f x 2 ( B x + z c ) ≥ ε C . L e t A ∈ F n × n 2 b e a matrix suc h that B = A + A T . Then ther e exist, for every y ∈ F n 2 , a ve ctor r y ∈ W such that E y ∈ W ∗ | E x ∈ y + W f ( x )( − 1) h x,Ax i + h B y ,x i + h r y ,x i | ≥ ε C . Pro of: Consider the quadratic phase g ( x ) = ( − 1) h x,Ax i and the linear p hase l ( z ) = ( − 1) h z ,z c i . (Note that th is is w here w e require B to hav e zero diagonal.) W e shall ﬁrst pr o v e that E x ∈ c 1 + W b f x 2 ( B x + z c ) = E x ∈ c 1 + W ( E y ∈ W ∗ h f x , g x l i y + W ) 2 ≤ E y ∈ W ∗ X α ∈ c W \ ( f g l ) y 2 ( α ) \ ( f g ) y 2 ( α ) , where again w e ha ve written h y ( x ) for the sh ift h ( x + y ) and the ﬁnal F ourier transform is tak en with resp ect to W . The equalit y follo ws fr om the fact that b f x ( B x + z c ) = E y f x ( y )( − 1) h y, B x + z c i = E y ∈ W ∗ E z ∈ y + W f x ( z )( − 1) h z ,B x + z c i and so ( − 1) h x,Ax i b f x ( B x + z c ) = E y ∈ W ∗ E z ∈ y + W f x ( z )( − 1) h z + x,A ( z + x ) i + h z ,Az i l ( z ) = E y ∈ W ∗ h f x , g x l i y + W , where the inn er pro duct is take n ov er the translate y + W . F or the inequalit y write E x ∈ c 1 + W ( E y ∈ W ∗ h f x , g x l i y + W ) 2 ≤ E y ∈ W ∗ E x ∈ c 1 + W h f x , g x l i 2 y + W , whic h equals E y ∈ W ∗ E x ∈ c 1 + W ( E z ∈ y + W f g l ( z ) f g ( z + x )) 2 = E y ∈ W ∗ E x ∈ W ( E z ∈ y + W f g l ( z ) f g ( z + x + c 1 )) 2 , whic h in turn can b e reexpressed as E y ∈ W ∗ E x ∈ W ( E z ∈ W ( f g l ) y ( z )( f g ) y ( z + x + c 1 )) 2 = E y ∈ W ∗ E x ∈ W (( f g l ) y ∗ W ( f g ) y )( x + c 1 ) 2 . T aking the F our ier transform with resp ect to W , it ca n b e seen that the latter expression equ als E y ∈ W ∗ X α ∈ c W \ ( f g l ) y 2 ( α ) \ ( f g ) y 2 ( α ) , completing the p ro of of the cla im from the b eginning. But since all functions inv olved are bou n ded, E y ∈ W ∗ X α ∈ c W \ ( f g l ) y 2 ( α ) \ ( f g ) y 2 ( α ) ≤ E y ∈ W ∗ sup α ∈ c W | \ ( f g ) y ( α ) | . No w for eac h y ∈ W ∗ , w e ﬁx a α y ∈ c W such that the supremum is attained. Then w e ha ve sh o wn that ε C ≤ E y ∈ W ∗ | \ ( f g ) y ( α y ) | = E y ∈ W ∗ | E x ∈ W f ( x + y )( − 1) h x + y , A ( x + y ) i + h α y ,x i | , whic h, after some rearranging of the phase, completes the pr o of. 33 5.5 Obtaining a qua dratic a verage Finally , w e use the sub s pace W from Section 5.4 to obtain the required quadratic av erage. Lemma 5.19 L et W 6 F n 2 b e a subsp ac e with co d ( V ) ≤ (1 /ε C ) . L et A ∈ F n × n 2 and B = A + A T b e such that ther e exist v e ctors r y ∈ W for e ach y ∈ W ∗ satisfying E y ∈ W ∗      E x ∈ y + W h f ( x )( − 1) h x,Ax i + h B y ,x i + h r y ,x i i      ≥ σ. Then for δ > 0 , one c an ﬁnd in time n 2 log n · | W ∗ | · p oly(1 /σ, log(1 /δ )) a quadr atic aver age with a ve ctor l y and a c onstant c y for e ach y ∈ W ∗ satisfying E y ∈ W ∗  E x ∈ y + W h f ( x )( − 1) h x,Ax i + h l y ,x i + c y i  ≥ σ 2 / 10 . Pro of: Let h y ( x ) def = f ( x )( − 1) h x, Ax i + h x,B y i . By assump tion we immediate ly ﬁnd that E y ∈ W ∗      E x ∈ y + W h h y ( x )( − 1) h r y ,x i i      = E y ∈ W ∗      E x ∈ W h h y y ( x )( − 1) h r y ,x i i      ≥ σ . Here h y y ( x ) = h y ( x + y ) as b efore. Without loss of generalit y , w e ma y assum e that the v ectors r y maximize the ab o v e expression. Thus, w e kn o w that on a v erage (o ve r y ), the functions h y y ha ve a large F ourier coeﬃcient (that is, signiﬁcant correlation with some v ector r y ∈ W ) o ver the subspace W . F or ev ery y ∈ W ∗ , we will use Theorem 5.16 to ﬁ nd th is F our ier coeﬃcient when it is indeed large. F or those y for w hic h the expression   E x ∈ W  h y y ( x )( − 1) h r y ,x i    is small for all r y ∈ W , w e will simply pic k an arbitrary p hase. Let us describ e this pr o cedure in more detail. Firs t, b y an a veraging argumen t w e kno w that E y ∈ W ∗      E x ∈ W h h y y ( x )( − 1) h r y ,x i i      ≥ σ ⇒ P y ∈ W ∗      E x ∈ W h h y y ( x )( − 1) h r y ,x i i     ≥ σ / 2  ≥ σ / 2 . Let S def = { y ∈ W ∗ :   E x ∈ W  h y y ( x )( − 1) h r y ,x i    ≥ σ / 2 } . Th e ab ov e inequalit y sho ws that | S | ≥ ( σ / 2) · W ∗ . No w for ea ch y ∈ W ∗ , we ru n the Goldreic h -Levin algorithm for the su bspace W fr om Theorem 5.16 w ith the f unction h y y , th e parameter γ = σ/ 2 and error probabilit y δ 2 / 2. F or eac h y ∈ S the algorithm ﬁnds, with probabilit y 1 − δ 2 , an r ′ y ∈ W and a c y ∈ F 2 satisfying E x ∈ W h h y y ( x )( − 1) h r ′ y ,x i + c y i ≥ σ / 4. Thus, with probabilit y 1 − δ / 2, it ﬁnds suc h an r ′ y for at least a 1 − δ f raction of y ∈ S . F or y / ∈ S , that is f or those y for whic h the algorithm fails to ﬁnd a goo d linear phase, w e c ho ose an r ′ y arbitrarily . If w e can force the co ntribution of terms for y / ∈ S to b e non-negativ e, then w e ha ve that with probabilit y 1 − δ / 2 E y ∈ W ∗  1 S ( y ) · E x ∈ W h h y y ( x )( − 1) h r ′ y ,x i + c y i  ≥ (1 − δ ) · ( σ / 2) · ( σ / 8) ≥ σ 2 / 9 . It remains to choose constan ts c y for y / ∈ S in suc h a w a y that their con tribution to the a verag e is non-negativ e. Consider the t wo p oten tial assignmen ts c y = 0 ∀ y / ∈ S and c y = 1 ∀ y / ∈ S . Clearly the con tribution of the terms for y / ∈ S m ust b e non-negativ e for at lea st one of the aforemen tioned assignmen ts, in whic h case w e obtain E y ∈ W ∗  E x ∈ W h h y y ( x )( − 1) h r ′ y ,x i + c y i  ≥ σ 2 / 9 . 34 In ord er to determine which of the tw o assignmen ts w orks, we can try b oth sets of signs and estimate the corresp onding quadratic av erage using O ((1 /σ 4 ) · log (1 /δ )) samples, and c ho ose the set of signs for which the estimate is larger. By Lemma 2.1 , with probabilit y at least 1 − δ / 2, w e select a set of v alues c y suc h that E y ∈ W ∗  E x ∈ y + W h f ( x )( − 1) h x,Ax i + h x,B y i + h x,r ′ y i + c y i  = E y ∈ W ∗  E x ∈ W h h y y ( x )( − 1) h r ′ y ,x i + c y i  ≥ σ 2 / 10 . Cho osing l y = B y + r ′ y then completes the pro of. 5.6 Putting things together W e no w giv e the pro of of Theorem 5.2. Pro of of T heorem 5.2: F or the pro cedure Find-Qua draticAv erage the fu nction ϕ ( x ) will b e sampled using Lemma 4.6 as r equired. W e start with a random u = ( x, ϕ ( x )) and a random c hoice of the parameters γ 1 , γ 2 , γ 3 as describ ed in the analysis of BS G-Test . W e also c h o ose the map Γ and the v alue c randomly for Mod el-Test . W e run the algorithm in Lemma 5.10 using BSG -Test and Model-Test with the ab o ve parameters, and with error parameter 1 / 4. Giv en a coset of the subs pace V and the map T , w e ﬁnd a subsp ace W ⊆ V and a symmetric matrix B with ze ro diagonal, usin g L emma 5.17. W e then use the algorithm in Lemma 5.1 9 to obtain the requ ired quadratic a verage , with probabilit y 1 / 4. Giv en a quadratic a v erage Q ( x ), w e estimate | h f , Q i| using O ((1 /σ 4 ) · log 2 ( θ /δ )) samples. If the estimate is le ss than σ 2 / 20, we discard Q and repeat the en tire pro cess. F or a M to be c h osen later, if we do not ﬁnd a qu adratic a verage in M attempts, w e stop and outp ut ⊥ . With probabilit y ρ/ 2, all samples of ϕ ( x ) (sampled with error 1 /n 5 ) corresp ond to a goo d fu nction ϕ . Conditioned on this, we h a v e a goo d choice of u and γ 1 , γ 2 , γ 3 for BSG-T est with probabilit y ρ 3 / 24. Also, we ha v e a go o d c hoice of the map Γ and c for Model-Te st w ith probability at least θ / 2 = ε O (1) . Conditioned on the ab o v e, the algorithm in Lemma 5.10 ﬁnds a go o d transf ormation with probabilit y 3 / 4 an d thus the output of the algorithm in Lemma 5.19 is a go o d quadratic a v erage with probability at least 1 / 2. Th u s, for M = O ((1 /ρ 4 ) · (1 /θ ) log (1 /δ )), the algorithm stops in M attempts with probabilit y at least 1 − δ/ 2. By c h oice of the num b er of samp les ab o ve , th e p robabilit y that w e estimate |h f , ( − 1) q i| incorrectly at an y step is at most δ/ 2 M . T herefore we output a go o d quadratic a verag e with probabilit y at least 1 − δ . The complexit y of the quadratic a verag e obtained, whic h is equal to the co-dimension of the sp ace W , is at O (1 /θ 3 ) = O (1 /ε C ). The run ning time of ea ch of the M ste p s is dominated b y that of the algorithm in Lemma 5.1 0 , whic h is O ( n 4 log 2 n · exp(1 /ε K )). W e conclud e that the total runn ing time is O ( n 4 log 2 n · exp(1 /ε K ) · log(1 /δ )). 6 Discussion One w a y in whic h one migh t w ant extend the results in this p ap er is to consid er the cyclic group of in tegers mo dulo of prime Z N . A (linear) Goldreich-Levin algorithm exists in this co ntext [A GS03], and some quadratic decomposition theorems ha v e b een prov en (see for example [GW10b]). Ho w- ev er, strong quan titativ e r esu lts in v olving the U 3 norm require a sig n iﬁcan t amount of eﬀort to ev en state. 35 F or example, the role of the su b space relativ e to wh ic h the quadratic a ve rages are deﬁned will b e pla y ed by so-called Bohr sets, whic h act as appro ximate subgroup s in Z N . Moreo ve r, it is no longer true that the in v erse theorem can guarant ee the exi stence of a g lobally deﬁned quadratic phase with wh ic h the function correlates; instead, this correlation ma y b e forced to b e (a nd remain) lo cal. Since there is an informal dictionary for translating analytic arguments from F n p to Z N , it see ms plausible th at many of our arguments could b e extended to this setting, at the co st of a d d ing a signiﬁcan t la y er of (largely tec hn ical) complexit y to the current presenta tion. 7 Ac kno wledgemen ts The auth ors w ould lik e to thank Tim Go wers, S w astik Koppart y , T om Sanders and Luca T r evisan for helpful con versatio n s . References [A GS03] Adi Ak a via, Shaﬁ Goldw asser, and Shm uel Safra, Pr oving har d-c or e pr e dic ates u sing list de c o ding , FOCS, 20 03, pp . 146–. [BS94] A. Balog and E. Szemer´ edi, A sta tistic al the or em of set additio n , Com bin atorica 14 (1994 ), 263– 268, 10.10 07/BF0121 2974. [BTZ10] V. Bergelson, T. T ao, and T . Ziegler, An inverse the or em for the u niformity se minorms asso ciate d with the action of F ω , Geom. F u nct. Anal. 16 (2010), no. 6, 1539 –1596. [BV10] And rej Bogdano v and Emanuele Viola, Pseu dor andom bits for p olynomials , SIAM J. Comput. 39 (20 10), n o. 6, 246 4–2486. [Can10] P . Candela, On the structur e of steps of thr e e-term arithmetic pr o g r essions in a dense set of inte gers , Bull. Lond. Math. So c. 42 (201 0), no. 1, 1–14. MR 2586 962 (2011a:110 17) [FK99] A. M. F rieze and R. Kannan, Qui c k app r oximation to matric es and ap plic ations , Com bi- natorica 19 (19 99), n o. 2, 175 –220. [GKZ08] P . Gopalan, A.R. Kliv an s , and D. Zuck erm an, List-de c o ding Re e d-Mul ler c o des over smal l ﬁelds , STOC, 200 8, pp . 265 –274. [GL89] O. Goldreic h and L. Levin, A har d-c or e pr e dic ate for al l one-way functions , Proceedings of the 21st ACM Sy m p osium on Theory of Computing, 198 9, p p. 25– 32. [Gop10] P . Gopalan, A Fourier-analytic appr o ach to Re e d-M u l ler de c o ding , F OCS, 2010 , pp. 685– 694. [Go w98] W.T. Go w ers, A new pr o of of Szemer´ edi’s the or em for arithmetic pr o gr essions of length four , Geom. F un c. An al. 8 (1998), no. 3, 529–551. [Gre07] B.J. Green, Montr ´ eal notes on quadr atic Fourier analysis , Additiv e com binatorics, CRM Pro c. Lecture Notes, vol. 43, Amer. Math. So c., Pro vidence, RI, 2007, pp. 69–102. MR 23594 69 (2008m:1 1047) 36 [GT08] B.J. Green and T. T ao, An inverse t he or em for the Gowers U 3 ( G ) norm , Pr o c. Edinb. Math. So c. (2) 51 (200 8), n o. 1, 73–153. MR 239163 5 (200 9g:11012) [GW10a ] W.T. Go wers and J. W olf, Line ar forms and qu adr atic uniformity fo r functions on F n p , T o app ear, Mathematik a. doi:10.1 112/S002557 9311001264, arXiv:10 02.2209 (20 10). [GW10b] , Line ar f orms and quadr atic uniformity for functions on Z N , T o app ear, J. Anal. Math., arXiv:1002 .2210 (2010 ). [GW10c ] , The true c omplexity of a system of line ar e quations , Proc. Lond. Math. S o c. (3) 100 (20 10), n o. 1, 155–176. MR 2578 471 (2011 a:11019) [HL11] Hamed Hatami and Shac har Lo ve tt, Corr elation testing for aﬃne invariant pr op erties on F n p in the high err or r e gime , 2011. [O’D08] R. O’Donnell, Some topics in analysis of Bo ole an functions , STOC , 2008, pp. 569– 578. [P et11] G. P etridis, Pl¨ unne cke’s Ine quality , Preprint, arXiv:1101.25 32 (2011). [Ruz99] I.Z. Ruzsa, An analo g of Fr eiman ’s the or em in gr oups , Ast ´ erisque (1999 ), no. 258, xv, 323–3 26, Structure theory of set addition. MR 170 1207 (2000h:1111 1) [Sam07] A. Samoro d nitsky , L ow-de gr e e tests at lar ge distanc es , Pro ceedings of the 39th ACM Symp osium on Theory of C omputing, 200 7, pp. 506–515. [SSV05] B. S u dak o v, E. Szemer´ edi, and V.H. V u , On a question of Er d¨ os and Moser , Duk e Mathematica l Jounal 129 (20 05), n o. 1, 129–155 . [ST06] Alex Samoro dnitsky and Luca T r evisan, Gowers unif ormity, inﬂuenc e of variables, and PCPs , STOC, 2006, p p . 11–20. [TTV09] L. T revisan, M. T u lsiani, and S . V adh an, Bo osting, r e gularity and e ﬃciently simulating every high-entr opy distribution , Pr o ceedings of the 24th IEEE Conference on Computa- tional Complexit y , 2009. [TV06] T. T ao and V. V u , A dditive c ombinatorics , Cam bridge Univ ersity Pr ess, 2006 . [TZ10] T. T ao and T. Ziegle r, The inverse c onje ctur e for the Gowers norm over ﬁnite ﬁelds via the c orr esp ondenc e principle , Analysis and PDE 3 (20 10), 1–20. [Vio07] E. Viola, Sele cte d r esu lts in add itive c ombinatorics: A n exp osition (pr eliminary version) , 2007. [VW07] Emanuele Viola and Avi Wigderson, Norms, XOR lemmas, and lower b ounds for GF(2) p olynomials and multip arty pr oto c ols , IEEE Conf erence on Computational Complexit y , 2007. 37

Quadratic Goldreich-Levin Theorems

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