The Lens of Abelian Embeddings

The Lens of Ab elian Em b eddings Dor Minzer Abstract. W e discuss a recen t line of research in v estigating inv erse theorems with respect to general k -wise correlations, and explain how suc h correlations arise in diﬀerent contexts in mathematics. W e outline some of the results that w ere established and their applications in discrete mathematics and theoretical computer science. W e also mention some op en problems for future research. 1 In tro duction. W e b egin the discussion about the study of k -wise correlations with motiv ating examples. 1.1 Arithmetic Progressions. Our ﬁrst example is the problem of upp er b ounding the size of sets of v ectors with no k -term arithmetic pr o gr essions . Definition 1.1. A k -term arithmetic pr o gr ession in F n p is a pr o gr ession of the form x, x + a, x + 2 a, . . . , x + ( k − 1) a , wher e x ∈ F n p and a ∈ F n p \ {  0 } . W e say a set A ⊂ F n p is a k -AP fr e e set if it c ontains no k -term arithmetic pr o gr essions. 1.1.1 Arithmetic Progressions of Length 3 . Using a densit y-incremen t approac h, Mesh ulam [50] (follo wing an argumen t of Roth [58]) used F ourier-analytic to ols to give an upp er b ound on the size of a set A ⊆ F n p that contains no 3 -term arithmetic progressions, where p is a prime. T o wards th is end Mesh ulam expresses the condition that A ⊆ F n p con tains no 3 -term arithmetic progressions analytically as E x,a ∈ F n p [1 A ( x )1 A ( x + a )1 A ( x + 2 a )] = µ ( A ) p − n , where µ ( A ) = | A | /p n is the relativ e size of A . T aking the normalized indicator f A ( x ) = 1 A ( x ) − µ ( A ) one concludes that if µ ( A ) ≥ 2 p − n/ 2 , then (1.1) E x,a ∈ F n p [ f A ( x ) f A ( x + a ) f A ( x + 2 a )] = − µ ( A ) 3 + µ ( A ) p − n ≤ − 1 2 µ ( A ) 3 , and in particular the left hand side has noticeable absolute v alue. What can be said ab out the structure of the function f A in that case, namely in the case that its v alues on the p oin ts x, x + a, x + 2 a has non-trivial correlation? T o answer this question we use the F ourier expansion of f A , whic h allows us to write f A ( x ) = P α ∈ F n p b f A ( α ) e 2 π i x · α , where x · α = n P i =1 x i α i and b f A ( α ) = E x ∈ F p [ f A ( x ) e − 2 π i x · α ] . An elementary calculation shows that E x,a ∈ F n p [ f A ( x ) f A ( x + a ) f A ( x + 2 a )] = X α ∈ F n p b f A ( α ) 2 b f A ( − 2 α ) , whic h quickly gives that there exists 0  = α ∈ F n p suc h that    b f A ( α )    ≥ Ω p ( µ ( A ) 2 ) . In words, f A m ust hav e a noticeable correlation with some F ourier c haracter. F rom this information one can deduce that there exists an aﬃne subspace W ⊆ F n p of co dimension 1 such that | A ∩ W | | W | ≥ µ ( A ) + Ω p ( µ ( A ) 2 ) . In words, the density of A inside W is signiﬁcantly larger than the o verall densit y of A . By iterating this argumen t, Meshulam even tually gets to a subspace W ′ of co dimension O p (1 /µ ( A )) in which the density of A is close to 1 . In that case A must clearly contain a 3 -term arithmetic progression. While the quan titative b ound obtained by Meshulam, which is µ ( A ) ≤ O p  1 n  , is b y no w obsolete thanks to the p olynomial method [21, 25], the F ourier-analytic approach underlying the argument plays a piv otal role in man y subsequent results in additiv e combinatorics, as discussed next. 1.1.2 Longer Progressions. The ﬁnite ﬁeld v ersion of Szemerédi’s theorem [61] asserts that for p > k , for any δ > 0 and suﬃciently large n ≥ n 0 ( δ, p ) , a set A ⊆ F n p with µ ( A ) ≥ δ m ust contain a k -term arithmetic progression. In other words, the measure of a set A ⊆ F n p with no k -term arithmetic progressions is a v anishing function of n . This result is a signiﬁcant extension of Meshulam’s result, and in its pro of Szemerédi in tro duces sev eral very impactful ideas suc h as the Szemerédi’s regularity lemma. Overall, Szemerédi’s argumen t is muc h more combinatorial in nature and has no resem blance to Meshulam’s argument at all. A dditionally , while the ideas used b y Szemerédi turn out to b e useful in a m uch broader context, they often hav e signiﬁcant drawbac ks, particularly with regard to the quan titative b ounds they giv e. F or example, the b ound on n 0 ( δ, p ) coming from Szemerédi’s argument is tow er-t yp e. Motiv ated by ﬁnding a higher order analog of Roth’s argument and by improving the b ounds on Szemerédi’s theorem, Gow ers [30] deﬁned the U s -uniformit y norms. Gow ers’ original deﬁnition works in the setting of the in tegers, and b elow we state the ﬁnite ﬁeld version. F or s ≥ 1 , the s -uniformity norm of a function f : F n p → C is deﬁned as ∥ f ∥ U s =   E x,h 1 ,...,h s ∈ F n p   Y w ∈{ 0 , 1 } s C | w | f ( x + s X i =1 w i h i )     1 / 2 s , where | w | = w 1 + . . . + w s and C j is the conjugate op erator for o dd j and the identit y for ev en j . Using these norms, Gow ers sho w ed that if A ⊆ F n p has noticeable density and the normalized indicator f A = 1 A − µ ( A ) has small ( k − 1) -uniformity norm, then A con tains many k -term arithmetic progressions (analogously to the case that k = 3 and f A has all small F ourier co eﬃcients). T o proceed by a densit y increment strategy , Gow ers then studies the case that f A has a noticeable uniformit y norm, whic h is signiﬁcan tly more c hallenging. Using a combination of com binatorial tec hniques and p ow erful to ols from additive combinatorics Gow ers shows a lo cal in verse theorem: there exists an aﬃne subspace W ⊆ F n p of co dimension O k,p (1) on which f A is correlated with a function of form e 2 π i ϕ ( x ) , where ϕ : F n p → F p is a degree- ( k − 2) p olynomial. This allows him to conclude that there exists a subspace W ′ ⊆ F n p of dimension n Ω k,p (1) in which A is noticeably denser. Since the work of Gow ers, the uniformity norms hav e b ecome an indisp ensable to ol in additive combinatorics and num ber theory , and their study has receiv ed m uch attention, see for example [5, 63, 36, 49, 55, 51, 47, 46]. 1.2 Constrain t Satisfaction Problems. Our next example comes from the ﬁeld of theoretical computer science, and more sp eciﬁcally from the area of hardness of approximation. 1.2.1 Decision and Optimization Problems. Definition 1.2. L et Σ b e a ﬁnite set, and let k ∈ N . A k -ary pr e dic ate over Σ is a map P : Σ k → { 0 , 1 } . W e often refer to the set Σ as the alphab et of the predicate, and to the parameter k as its arity . With any collection P of k -ary predicates ov er Σ we asso ciate the computational problem P -CSP , deﬁned as follows. Definition 1.3. F or a c ol le ction of P of k -ary pr e dic ates over Σ , an instanc e I of P -CSP c onsists of a set of variables X = { x 1 , . . . , x n } and a set of c onstr aints E = { e 1 , . . . , e m } . Each c onstr aint e i c onsists of k -variables x i 1 , . . . , x i k ∈ X and a pr e dic ate P i ∈ P , and it is thought of as imp osing the c ondition P i ( x i 1 , . . . , x i k ) = 1 . Let P b e a collection of predicates, and let I b e an instance of P -CSP . An assignment to I is a labeling A : X → Σ . W e sa y an assignment A satisﬁes a constrain t e i giv en as P i ( x i 1 , . . . , x i k ) = 1 , if P i ( A ( x i 1 ) , . . . , A ( x i k )) = 1 . W e deﬁne the v alue of the assignment A as val I ( A ) = |{ e i ∈ E | A satisﬁes e i }| | E | , and deﬁne the v alue of an instance I as val ( I ) = max A : V → Σ val I ( A ) . W e consider the following computational problems: 1. Decision-Satisﬁabilit y: given an instance I of P -CSP , determine if val ( I ) = 1 or not. In words, design an algorithm that given an instance I of P -CSP , accepts if val ( I ) = 1 , and rejects otherwise. 2. Gap-Maximization: for n umbers 0 ≤ s ≤ c ≤ 1 , design an algorithm that given an instance I of P -CSP promised to either ha ve val ( I ) ≥ c or val ( I ) < s , accepts in the former case and rejects in the latter case. W e often denote this promise problem by gap- P -CSP [ c, s ] . Problems as in the ﬁrst item ab ov e, often referred to as decision problems in the literature, are the basis of the theory of NP-completeness [20, 48, 42]. While the classical theory of NP-completeness go es far b ey ond the scop e of constrain t satisfaction problems, m uch eﬀort has gone in to c haracterizing collections P when the ab o ve task is computationally feasible. The dichotom y theorem of Zh uk and Bulato v [66, 19] (which for a long time w as a prominen t outstanding conjecture) states that for any collection of predicates P , the decision problem asso ciated with P -CSP can either b e solved in p olynomial time, or else is NP-hard. Problems as in the second item ab ov e, often referred to as gap problems or optimization problems, are the basis of the theory of NP-hardness of approximation problems. This is precisely the type of problems addressed b y the PCP theorem [26, 3, 2], whic h in these terms asserts that for Σ = { 0 , 1 } and k = 3 , there is s < 1 and a collection of 3 -ary predicates P suc h that gap- P -CSP [1 , s ] is NP-hard. While the theory of approximation is b y no w well-dev elop ed, we are still far from a complete understanding of the complexit y of approximation of CSPs. 1.2.2 The Almost Satisﬁable Regime. The problem gap- P -CSP [ c, s ] turns out to exhibit a very diﬀerent b eha vior dep ending on whether c = 1 (the “satisﬁable regime”), or c < 1 (typically c = 1 − ε for a small ε > 0 , the “almost satisﬁable regime”). T o demonstrate this w e consider the 3 -Lin problem. An instance of the 3 -Lin problem I consists of a collection of v ariables X = { x 1 , . . . , x n } that are supp osed to b e assigned v alues from F p , and a collection of equations of the form ax i + bx j + cx k = d where x i , x j , x k ∈ X and a, b, c, d ∈ F p are constants (it is easy to formalize this problem as a CSP as deﬁned abov e). The goal is to ﬁnd an assignment A : X → F p satisfying as many of the equations as p ossible. F or this problem, it is easy to see that gap- 3 -Lin [1 , s ] can b e solved in p olynomial time for an y s ≤ 1 . Indeed, the Gaussian elimination algorithm from linear algebra can b e used to determine if the giv en system of linear equations has a solution, i.e. if val ( I ) = 1 , or not. The situation is completely diﬀerent for c < 1 , and the Gaussian elimination algorithm no longer works. It can b e shown that a randomly chosen assignment satisﬁes in exp ectation 1 /p fraction of the equations, so it is alw ays the case that val ( I ) ≥ 1 /p . Th us, gap- 3 -Lin [ c, 1 /p ] can b e solved in p olynomial time for all c < 1 . It turns out that this is essentially the b est one can do: Theorem 1.4 (Håstad [39]). F or any prime p and ε > 0 , the pr oblem gap- 3 -Lin [1 − ε, 1 /p + ε ] is NP-har d. Håstad’s pro of of Theorem 1.4 proceeds b y a reduction from the PCP theorem, and in his analysis he uses discrete F ourier analysis (in fact, he uses a similar F ourier-analytic argument to the one in Subsection 1.1.1). F ourier-analytic to ols hav e since b ecome ubiquitous in the area. Subsequen t researc h led to an almost complete understanding of the “almost satisﬁable regime”, at least assuming a complexit y theoretic assumption known as the Unique-Games Conjecture [43] (which will not be discussed here; see [44, 64, 4, 45, 52] for more information). This culminated in a result b y Raghav endra [56], who prov ed the following dic hotomy-t ype result: Theorem 1.5 (Ragha v endra). F or any ﬁnite alphab et Σ , k ∈ N , a c ol le ction of k -ary pr e dic ates P over Σ , c ∈ [0 , 1] and ε > 0 , ther e exists s ∈ [0 , 1] such that t he fol lowing holds: 1. A lgorithm: ther e is a p olynomial time algorithm solving gap- P -CSP [ c − ε, s ] . 2. Har dness: assuming the Unique-Games c onje ctur e, for al l δ > 0 gap- P -CSP [ c − ε, s + δ ] is NP-har d. 1.2.3 Dictatorship T ests. The proof of Theorem 1.5 also pro ceeds b y a reduction and uses discrete F ourier analysis. A key comp onent in this result is a dictatorship test . Roughly sp eaking, a ( c, s ) -dictatorship test for a predicate P : Σ k → { 0 , 1 } is a distribution µ ov er Σ k suc h that the following holds for large enough n : 1. Completeness: if f : Σ n → Σ is a dictatorship function, namely , it is a function of the form f ( x ) = x i for some i ∈ { 1 , . . . , n } , then E ( x 1 ,...,x k ) ∼ µ ⊗ n [ P ( f ( x 1 ) , . . . , f ( x k ))] ≥ c. Here, the distribution µ ⊗ n is the distribution ov er (Σ n ) k where for each i ∈ { 1 , . . . , n } , ( x 1 ( i ) , . . . , x k ( i )) is sampled indep enden tly according to µ . 2. Soundness: if f : Σ n → Σ is a function that is quasi-random with resp ect to dictatorships, then E ( x 1 ,...,x k ) ∼ µ ⊗ n [ P ( f ( x 1 ) , . . . , f ( x k ))] ≤ s + o (1) . By “quasi-random with resp ect to dictatorships” we mean that for each a ∈ Σ , the function f a : Σ n → { 0 , 1 } deﬁned as f a ( x ) = 1 f ( x )= a has that the individual inﬂuences I i [ f a ; µ ⊗ n r ] = Pr x,x ′ ∼ µ ⊗ n r [ f a ( x )  = f a ( x ′ ) | x j = x ′ j ∀ j  = i ] are o (1) for all i = 1 , . . . , n and r = 1 , . . . , k , where µ r is the marginal distribution of µ on co ordinate r . In the pro of of Theorem 1.5 R agha v endra gives an appro ximation algorithm based on semi-deﬁnite programming. He then considers in tegrality gaps for this algorithm – namely instances which are “hardest” for the algorithm, and sho ws how to construct a dictatorship test µ with parameters nearly matching the p erformance of the algorithm. A subtle point that initially seems v ery minor is that Raghav endra m ust ensure that the support of the distribution µ , denoted by supp ( µ ) , is the entiret y of Σ k . Ultimately , this leads to the completeness parameter b eing c − ε (instead of c ) in b oth of the items in Theorem 1.5. This fact is necessary in the analysis of the dictatorship test µ that he constructs. Roughly sp eaking, after arithmetizing the predicate P , the analysis of the dictatorship tests b oils down to understanding exp ectations of the form (1.2) E ( x 1 ,...,x k ) ∼ µ ⊗ n [ f 1 ( x 1 ) · · · f k ( x k )] for some 1 -b ounded functions f 1 , . . . , f k arising from f (in fact, each f i is f a i for some a i ∈ Σ ). Raghav endra argues that almost all the contribution to this expectation comes from the low-degree F ourier part of the functions f i , at whic h p oin t he can replace the f i with their low-degree part. The argument is then ﬁnished by app ealing to the inv ariance principle of Mossel, O’Donnell and Oleszkiewicz [54], which relates the correlation of the low-degree parts to a similar exp ectation in Gaussian space. T o argue that essentially all contribution to this exp ectation comes from the low-degree parts of the functions Ragha vendra uses a result of Mossel [53], whic h is the place where the fact that supp ( µ ) = Σ k is used crucially . 1.2.4 The Satisﬁable Regime. Results in the satisﬁable regime are muc h more rare in the literature, and we are far from having a complete understanding of the complexity of gap- P -CSP [1 , s ] for general P . In fact, this is well understo od only for very few predicates. An example is the 3 -SA T problem, which is deﬁned b y the 8 predicates { P a,b,c : { 0 , 1 } 3 → { 0 , 1 } | a, b, c ∈ { 0 , 1 }} given as P a,b,c ( x, y , z ) = 1 − 1 x = a,y = b,z = c . It is shown in [39] that gap- 3 -SA T [1 , s ] can b e solved in polynomial time for s = 7 8 and is NP-hard for any s ≥ 7 8 + ε . The satisﬁable regime remains rather po orly understoo d ev en if one is willing to assume conjectures in the spirit of the Unique-Games Conjecture, such as the Rich 2 -to- 1 Games Conjecture [17]. This is because essen tially all hardness of approximation results are prov ed by constructing a dictatorship test as ab o ve (as is the case in Theorem 1.5), and to address the satisﬁable regime one m ust come up with dictatorship tests with c = 1 . In particular, the supp ort of µ must b e con tained in P − 1 (1) , and w e can no longer aﬀord to mo dify the distribution µ so that supp ( µ ) = Σ k en tirely . W e must work with µ as it is. 2 The Main Analytical Questions. The former discussion naturally leads to the following question: Question 2.1. F or what distributions µ over Σ k is the fol lowing assertion true: for al l 1 -b ounde d functions f 1 , . . . , f k : Σ n → C such that at le ast one of the f i is essential ly only on de gr e es higher than d , we have that   E ( x 1 ,...,x k ) ∼ µ ⊗ n [ f 1 ( x 1 ) · · · f k ( x k )]   → d →∞ 0 . Here and throughout, w e will often use the somewhat informal term “essen tially on degrees higher than d ”. By that, we mean that if we expand f i according to its Efron-Stein decomp osition, only a very small amount of mass remains on degrees low er than d . A t ypical example is a function of the form f i = (I − T 1 − ε/d ) g i , where g i is a 1 -b ounded function, I is the identit y op erator, and T 1 − ε/d is the standard noise op erator, deﬁned as follows. Definition 2.2. F or a distribution ν over Σ and a p oint x ∈ Σ n , a sample fr om the distribution T 1 − ε,ν x is dr awn by taking for e ach i ∈ { 1 , . . . , n } indep endently y i = x i with pr ob ability 1 − ε , and else sampling y i ∼ ν . When ν is cle ar fr om c ontext we often omit it fr om notation. Abusing notations, we also think of T 1 − ε as an aver aging op er ator, wher e for f : Σ n → C we deﬁne T 1 − ε f ( x ) = E y ∼ T 1 − ε x [ f ( y )] . As discussed earlier, Mossel [53] prov ed that the assertion of Question 2.1 holds if supp ( µ ) = Σ k . In fact, he show ed that it is suﬃces that supp ( µ ) is c onne cte d . By that, w e mean that the graph G µ whose vertices are supp ( µ ) , and ( a 1 , . . . , a k ) , ( b 1 , . . . , b k ) ∈ supp ( µ ) are adjacent if they diﬀer in a single c oordinate, is connected. 2.1 The Case of No Ab elian Embeddings. In an eﬀort to mak e progress on the complexit y of appro ximation of CSPs in the satisﬁable regime, the authors of [14] hypothesize an answ er to Question 2.1. T o state it we require the notion of Ab elian emb e ddings , whic h is central to the discussion. Definition 2.3. L et k ∈ N , let Σ 1 , . . . , Σ k b e ﬁnite alphab ets and let µ b e a distribution over Σ 1 × . . . × Σ k such that for e ach i , supp ( µ i ) = Σ i . W e say that µ admits an Ab elian emb e dding if ther e is an A b elian gr oup ( A, +) and maps σ i : Σ i → A not al l c onstant such that σ 1 ( x 1 ) + . . . + σ k ( x k ) = 0 A for al l ( x 1 , . . . , x k ) ∈ supp ( µ ) . In words, a distribution is said to admit an Ab elian embedding if the alphab et symbols can b e (non-trivially) lab eled by elemen ts of an Ab elian group ( A, +) , such that under this lab eling, supp ( µ ) is con tained in the solution space of some linear equation o ver A . The hypothesis of [14] reads: Hyp othesis 2.4. Supp ose that µ is a distribution ov er Σ k that do es not admit any Ab elian embedding. Then the assertion of Question 2.1 holds. W e remark that if µ admits an Ab elian embedding, then the assertion of Question 2.1 fails. Indeed, take an Ab elian group ( A, +) and maps σ 1 , . . . , σ k as in Deﬁnition 2.3, and choose non-trivial characters χ 1 , . . . χ n ∈ ˆ A . Deﬁne the functions f 1 , . . . , f k as f i ( x i ) = n Q j =1 χ j ( σ i ( x i ( j ))) . Then for all ( x 1 , . . . , x k ) ∈ supp ( µ ⊗ n ) we hav e k Y i =1 f i ( x i ) = k Y i =1 n Y j =1 χ j ( σ i ( x i ( j ))) = n Y j =1 k Y i =1 χ j ( σ i ( x i ( j ))) = n Y j =1 χ j ( k X i =1 σ i ( x i ( j ))) = n Y j =1 χ j (0 A ) = 1 , and in particular the exp ectation of k Q i =1 f i ( x i ) ov er µ ⊗ n has a large absolute v alue. Also, taking i such that σ i is not constan t, it can b e chec k ed that almost all of the mass of f i is on degrees Θ( n ) . This example means that, if true, Hyp othesis 2.4 would give a complete answer to Question 2.1. 2.2 The Case of A belian Em b eddings. Recall that in Section 1.1 we discussed k -wise correlations of the form (1.2) where the distribution µ is uniform ov er ( x, x + a, x + 2 a, . . . , x + ( k − 1) a ) ov er all x, a ∈ F p . Observ e that this µ do es admit Ab elian embeddings; for example, one can tak e σ i : F p → F p deﬁned as σ 1 ( x ) = x , σ 2 ( y ) = − 2 y , σ 3 ( z ) = z and σ i ≡ 0 for i > 3 . Th us, Hyp othesis 2.4 do es not apply for µ . This raises the question: what can we say about the inv erse problem for correlations such as (1.2) when µ do es hav e Ab elian embeddings? A quic k insp ection sho ws that for the problem to hav e any meaningful answer we m ust mak e some mild assumption on µ . F or instance, if µ is only supported on tuples of the form ( x, . . . , x ) ∈ Σ k , then essentially nothing of in terest can b e said ab out functions f 1 , . . . , f k that ac hieve high k -wise correlation with resp ect to µ . A natural, fairly mild, prop erty for µ turns out to b e p airwise-c onne cte dness . Here and throughout, for i, j ∈ { 1 , . . . , k } , µ i and µ i,j are the marginal distributions of µ on co ordinate i , and on coordinates i, j , resp ectiv ely . Definition 2.5. L et k ∈ N and let Σ 1 , . . . , Σ k b e ﬁnite alphab ets. W e say that a distribution µ over Σ 1 × . . . × Σ k is p airwise-c onne cte d if any distinct i, j ∈ { 1 , . . . , k } , the bip artite gr aph G i,j , whose sides ar e Σ i , Σ j and whose e dge set is supp ( µ i,j ) , is c onne cte d. It can b e chec ked that for a distribution µ , the follo wing chain of implications holds: µ is connected = ⇒ µ has no Ab elian embeddings = ⇒ µ is pairwise-connected , and so among the 3 notions, being pairwise-connected is the mildest property . It can be seen that all of the distributions w e considered so far (including those arising in the con text of arithmetic progressions) are pairwise-connected. The following question asks for a char acterization of 1 -b ounded functions f 1 , . . . , f k ac hieving noticeable correlation with resp ect to the k -wise correlation deﬁned b y µ , in the case it is pairwise-connected: Question 2.6. L et µ b e a p airwise-c onne cte d distribution over Σ k . Supp ose f 1 , . . . , f k : Σ n → C ar e 1 -b ounde d functions such that   E ( x 1 ,...,x k ) ∼ µ ⊗ n [ f 1 ( x 1 ) · · · f k ( x k )]   ≥ ε, wher e ε > 0 . What c an we say ab out the structur e of the functions f i ? Belo w, after discussing some results, we state a natural candidate for the structure of functions as in Question 2.6. W e remark that the setting of Question 2.6 generalizes the setting of Question 2.1, so the answ er must include the p ossibilit y that f i are correlated with low-degree functions. F or k = 3 , the setting of Question 2.6 includes within it the setting of arithmetic progressions of length 3 , so the answer m ust include the p ossibilit y that f i are correlated with some F ourier character coming from an Ab elian group (which is not clear a priori, since now Σ is a completely abstract set, void of an y algebraic structure). More generally , for large k = 2 s the setting of Question 2.6 includes within it k -term arithmetic progressions as well as U s uniformit y norms. Th us, the answer m ust include the p ossibilit y that f i is correlated with functions arising in the inv erse theorem for Go wers uniformity norms as in [62]. As the inv erse theorem for Gow ers uniformity norms gets signiﬁcantly more c hallenging as s increases, it is natural to exp ect that Question 2.6 gets more challenging as k increases. 3 In v erse Theorems for k -Wise Correlations: Statemen ts. In this section we discuss results tow ards the resolution of Question 2.1 and Question 2.6, and in Section 4 we giv e an informal description of the pro ofs. 3.1 The Case of 3 -ary Distributions with No Ab elian Embeddings. The pap er [14] raised Hyp oth- esis 2.4 and prov ed it is correct in the case k = 3 and the distribution µ is a union of matchings . By that, w e mean that for each x , there is a p erfect matching M x : Σ → Σ such that supp ( µ ) = S x ∈ Σ { ( x, y , M x ( y )) | y ∈ Σ } . Their result reads as follows: Theorem 3.1. F or al l α, ε > 0 ther e exist δ > 0 and d ∈ N such that the fol lowing holds. Supp ose µ is a distribution over Σ 3 satisfying: 1. The pr ob ability of e ach atom in µ is at le ast α . 2. The distribution µ has no Ab elian emb e ddings and is a union of matchings. Then if f , g , h : Σ n → C ar e 1 -b ounde d functions satisfying   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then ther e exists a function L : (Σ n , µ ⊗ n 1 ) → C of de gr e e at most d and 2 -norm e qual to 1 such that |⟨ f , L ⟩| ≥ δ . While there is nothing inherently interesting ab out the “union of matchings” condition, it is a natural case to consider from a technical p oin t of view. The pro of of Theorem 3.1 in volv es a certain pro cedure called “the path tric k” that, in a sense, enriches the distribution µ . The case of union of matchings can b e seen as sort of limiting p oin t of that pro cess. Theorem 3.1 has b een subsequently impro ved in [15], who remov ed the “union of matchings” condition. Their result reads as follows: Theorem 3.2. F or al l α, ε > 0 ther e exist δ > 0 and d ∈ N such that the fol lowing holds. Supp ose µ is a distribution over Σ 3 satisfying: 1. The pr ob ability of e ach atom in µ is at le ast α . 2. The distribution µ has no Ab elian emb e ddings. Then if f , g , h : Σ n → C ar e 1 -b ounde d functions satisfying   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then ther e exists a function L : (Σ n , µ ⊗ n 1 ) → C of de gr e e at most d and 2 -norm e qual to 1 such that |⟨ f , L ⟩| ≥ δ . The pro of of Theorem 3.2 uses some of the comp onents of the pro of of Theorem 3.1, but do es not reduce to the case therein. Instead, the pro of pro ceeds b y induction on n , the n um b er of co ordinates, via a sort of tensorization argument. Theorem 3.2 resolves Hyp othesis 2.4 for k = 3 , and it is natural to next consider the case of k = 4 . This case turns out to be signiﬁcantly more diﬃcult, and we do not know how to reduce it to the k = 3 case. A natural attempt at such a reduction pro ceeds as follows: supp ose w e hav e 1 -b ounded functions f 1 , . . . , f 4 : Σ n → C , and let µ b e a distribution ov er Σ 4 . Consider the distribution µ ′ o ver (Σ ′ ) 3 where Σ ′ = Σ 2 , deﬁned as follows: 1. Sample ( x, y , z , w ) ∼ µ . 2. Sample ( x ′ , y ′ , z ′ , w ′ ) ∼ µ conditioned on w ′ = w . 3. Output (( x, x ′ ) , ( y , y ′ ) , ( z , z ′ )) . With this distribution in mind, one may use the 1 -b oundedness of f 4 and Cauch y-Sc hw arz to get that   E ( x,y ,z,w ) ∼ µ ⊗ n [ f 1 ( x ) f 2 ( y ) f 3 ( z ) f 4 ( w )]   2 =    E w ∼ µ ⊗ n 4 [ f 4 ( w ) E ( x,y ,z,w ′ ) ∼ µ ⊗ n [ f 1 ( x ) f 2 ( y ) f 3 ( z ) | w ′ = w ]]    2 ≤ E w ∼ µ ⊗ n 4 [   E ( x,y ,z,w ′ ) ∼ µ ⊗ n [ f 1 ( x ) f 2 ( y ) f 3 ( z ) | w ′ = w ]   2 ] = E (( x,x ′ ) , ( y ,y ′ ) , ( z ,z ′ )) ∼ µ ′ ⊗ n [ F 1 ( x, x ′ ) F 2 ( y , y ′ ) F 3 ( z , z ′ )] , (3.1) where F 1 ( x, x ′ ) = f 1 ( x ) f 1 ( x ′ ) , F 2 ( y , y ′ ) = f 2 ( y ) f 2 ( y ′ ) and F 3 ( z , z ′ ) = f 3 ( z ) f 3 ( z ′ ) . It can b e sho wn that if f i is high-degree for some i ∈ { 1 , 2 , 3 } , then the same holds for F i . Th us, the abov e inequality w ould b e a reduction from the k = 4 to the k = 3 case had it b een the case that µ ′ has no Abelian embeddings whenever µ do es not. Alas, this turns out to b e false. Ev en though the ab o v e candidate reduction fails, it is not completely useless. W e note that the supp ort of µ ′ con tains within it a copy of µ , namely the “equal pairs” tuples (( x, x ) , ( y , y ) , ( z , z )) for all ( x, y , z ) ∈ supp ( µ ) , and furthermore these inputs hav e a noticeable mass α ′ (whic h is at least α 2 ) in µ ′ . Th us, to generate a sample according to µ ′ , one could write µ ′ = α ′ µ ′′ + (1 − α ′ ) µ ′′′ where µ ′′ is the distribution µ conditioned on b eing on equal pairs, and µ ′′′ is the distribution µ conditioned on b eing on non-equal pairs, and then with probabilit y α ′ sample according to µ ′′ , and with probabilit y 1 − α ′ sample according to µ ′′′ . T o sample according to µ ⊗ n , one could sample I ⊆ α ′ [ n ] , b y which we mean w e include each i ∈ [ n ] in I with probabilit y α ′ indep enden tly , then sample the coordinates of I as (( x, x ) , ( y , y ) , ( z , z )) ∼ µ ′′ I and the coordinates of ¯ I as (( x ′ , ˜ x ) , ( y ′ , ˜ y ) , ( z ′ , ˜ z )) ∼ µ ′′′ ¯ I . With that in mind one can write the right hand side of (3.1) as E I ⊆ α ′ [ n ] (( x ′ , ˜ x ) , ( y ′ , ˜ y ) , ( z ′ , ˜ z )) ∼ µ ′′′ ¯ I " E (( x,x ) , ( y ,y ) , ( z ,z )) ∼ µ ′′ I [( F 1 ) ¯ I → ( x ′ , ˜ x ) ( x, x )( F 2 ) ¯ I → ( y ′ , ˜ y ) ( y , y )( F 3 ) ¯ I → ( z ′ , ˜ z ) ( z , z )] # , where ( F 1 ) ¯ I → ( x ′ , ˜ x ) : Σ ′ I → C is the function F 1 where the co ordinates of ¯ I hav e b een ﬁxed to ( x ′ , ˜ x ) , and similarly for F 2 and F 3 . The inner exp ectation can now b e thought of as an exp ectation ov er µ , so one could deduce from the k = 3 case that ( F 1 ) ¯ I → ( x ′ , ˜ x ) is correlated with a lo w-degree function for man y restrictions ¯ I , ( x ′ , ˜ x ) . What could we say ab out F 1 is that case? It is useful to consider the case the distribution µ is supp orted on arithmetic progressions. In that case F 1 , F 2 , F 3 could b e viewed as multiplicativ e deriv atives of f 1 , f 2 , f 3 , and the information we gathered indicates that they tend to b e low-degree functions under axis-parallel restrictions. Morally , low-degree functions could b e though t of as constant functions. Thus, as functions with constan t multiplicativ e deriv atives must b e a F ourier c haracters, one exp ects something similar to hold for F 1 . A t the presen t setting though, this last assertion do es not mak e complete sense. F or once, w e do not ev en ha ve a group in the setting of a general µ as ab o v e. This naturally leads to Question 2.6, with the hop e that the answ er therein will b e useful in the ab ov e attempt for the k = 4 case (which is indeed the case). 3.2 The Case of 3 -ary Distributions with Ab elian Em b eddings. W e now discuss results regard- ing Question 2.6 in the case that k = 3 . There is a signiﬁcant diﬀerence dep ending on if the distribution µ admits an Ab elian embedding into an inﬁnite group or not, and we use the following deﬁnition: Definition 3.3. W e say a distribution µ over Σ 1 × . . . Σ k admits ( Z , +) -emb e ddings if ther e ar e maps σ 1 , . . . , σ k : Σ → Z not al l c onstant such that σ 1 ( x 1 ) + . . . + σ k ( x k ) = 0 for al l ( x 1 , . . . , x k ) ∈ supp ( µ ) . W e note that whenev er a distribution µ admits a ( Z , +) -em b eddings, it also admits em b eddings in to arbitrarily large ﬁnite Ab elian groups (taking σ i mo dulo p for suﬃciently large p , for example). The reverse is not true, though, and there are distributions that admit Ab elian embeddings but hav e no ( Z , +) -embeddings. F or example, the uniform distribution ov er { ( x, x + a, x + 2 a ) | x ∈ F p , a ∈ { 0 , 1 , 2 }} admits Ab elian embeddings but not ( Z , +) -em b eddings, whereas the uniform distribution ov er { ( x, x + a, x + 2 a ) | x ∈ F p , a ∈ { 0 , 1 }} admits ( Z , +) - em b eddings. T o appreciate the diﬀerence b et ween the case that µ has ( Z , +) -embeddings and the case it do es not, it is useful to insp ect the recip e given b elow Hyp othesis 2.4 for functions achieving noticeable 3 -correlations. Therein one could take ( A, +) to b e any group which µ embeds into. If µ has no ( Z , +) -embeddings then there are only ﬁnitely man y suc h A ’s, and the functions f 1 , . . . , f k are discrete v alued. On the other hand, if ( A, +) = ( Z , +) then one can pick χ j to b e any function of the form χ j ( a ) = e 2 π i θ j a for θ j ∈ (0 , 1) . In particular, the choice of θ j could dep end on the num b er of v ariables n (for example, it could b e 1 √ n ). This is a signiﬁcantly richer class of examples, and they cannot b e attributed to a single group of a ﬁxed, constant size indep endent of n . 3.2.1 The Sub case of No ( Z , +) -em b eddings. The pap er [13] considered Question 2.6 in the case that k = 3 and the distribution µ has no ( Z , +) -embeddings. Their main technical result is a lo cal inv erse theorem: Theorem 3.4. F or al l m ∈ N ther e is a gr oup G of size O m (1) such that for al l ε, α > 0 ther e exists δ > 0 , such that if µ is a distribution over Σ 3 in which | Σ | = m , the pr ob ability of e ach atom is at le ast α and µ has no ( Z , +) -emb e ddings, then ther e is a map σ : Σ → G such that the fol lowing holds. If f , g , h : Σ n → C ar e 1 -b ounde d functions such that   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then Pr I ⊆ δ [ n ] ˜ x ∼ µ ¯ I 1 h ∃ χ ∈ b G I such that   ⟨ f ¯ I → ˜ x , χ ◦ σ ⊗ I ⟩   ≥ δ i ≥ δ. Her e, f ¯ I → ˜ x : Σ I → C is the function f when we r estrict the c o or dinates of ¯ I ac c or ding to ˜ x , and σ ⊗ I : Σ I → G I is deﬁne d as σ ⊗ I ( x ) = ( σ ( x i )) i ∈ I . In words, Theorem 3.4 asserts that if f , g , h achiev e noticeable 3 -wise correlation according to a distribution µ with no ( Z , +) -embeddings, then after randomly restricting all but δ fraction of the co ordinates, it is correlated with a character arising from some ﬁnite group G of a ﬁxed size. T o get a global inv erse theorem they establish a “restriction in verse theorems” [11, 13], giving structural information ab out a function f from structural information ab out it under random restrictions: Theorem 3.5. Supp ose G is a ﬁnite Ab elian gr oup, σ : Σ → G is a map and ν is a distribution over Σ in which the pr ob ability of e ach atom is at le ast α . Then for al l ε > 0 ther e ar e d ∈ N and δ > 0 such that if f : Σ n → C is a 1 -b ounde d function satisfying Pr I ⊆ ε [ n ] ˜ x ∼ ν ¯ I h ∃ χ ∈ b G I such that   ⟨ f ¯ I → ˜ x , χ ◦ σ ⊗ I ⟩   ≥ ε i ≥ ε, then ther e exist χ ∈ b G n and L : (Σ n , ν ⊗ n ) → C with ∥ L ∥ 2 = 1 and deg ( L ) ≤ d such that |⟨ f , L · χ ◦ σ ⊗ n ⟩| ≥ δ . Com bining Theorems 3.4 and 3.5 gives an answer to Question 2.6 in the case k = 3 and µ has no ( Z , +) -embeddings: Theorem 3.6. In the setting of The or em 3.4 , for al l ε > 0 ther e ar e d ∈ N and δ > 0 such that if f , g , h : Σ n → C ar e 1 -b ounde d functions satisfying   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then ther e exist χ ∈ b G n and L : (Σ n , µ ⊗ n 1 ) → C with ∥ L ∥ 2 = 1 and deg ( L ) ≤ d such that |⟨ f , L · χ ◦ σ ⊗ n ⟩| ≥ δ . It is not clear though how to generalize the arguments of [13] to the case µ has ( Z , +) -em b eddings. The main issue is that the arguments in [13] heavily rely on the fact that group G can b e identiﬁed by only lo oking at the distribution µ . In particular, it cannot dep end on the num ber of co ordinates n . 3.2.2 The Pairwise-connected Case. The statements of Theorems 3.4 to 3.6 still mak e some sense for 3 -ary , pairwise-connected distributions µ (namely , besides the fact that the size of G dep ends only on m ). In particular, one is naturally led to the follo wing candidate statemen t: in the setting of Theorem 3.4, if µ is only assumed to b e pairwise-connected (as opp osed to having no ( Z , +) -embeddings), then after random restriction it is correlated with some character function coming from ( Z , +) . W e no w mak e a few remarks. First, this candidate statement is false if we do not allo w for random restrictions. In fact, w e are not aw are of any similar plausible statemen t that do es not in volv e random restrictions, and this complicates matters. Second, once w e allo w ( Z , +) -characters, the group structure becomes more of a red-herring, and it is b etter to abstract out this structure in the language of pr o duct functions . Definition 3.7. W e say a function P : Σ n → C is a pr o duct function if ther e ar e univariate functions P 1 , . . . , P n : Σ → C such that P ( x 1 , . . . , x n ) = P 1 ( x 1 ) · · · P n ( x n ) for al l x 1 , . . . , x n ∈ Σ . With this deﬁnition, the main result of [8] is the follo wing lo cal inv erse theorem: Theorem 3.8. F or al l ε, α > 0 ther e exists δ > 0 , such that the fol lowing holds for any p airwise-c onne cte d distribution µ over Σ 3 in which the pr ob ability of e ach atom is at le ast α . If f , g , h : Σ n → C ar e 1 -b ounde d functions such that   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then Pr I ⊆ δ [ n ] ˜ x ∼ µ ¯ I 1  ∃ P : Σ I → C a pr o duct function with ∥ P ∥ 2 ≤ 1 such that |⟨ f ¯ I → ˜ x , P ⟩| ≥ δ  ≥ δ. The pap er [8] also prov es a version of Theorem 3.5 for pro duct functions: Theorem 3.9. Supp ose ν is a distribution over Σ in which the pr ob ability of e ach atom is at le ast α . Then for al l ε > 0 ther e ar e d ∈ N and δ > 0 such that if f : Σ n → C is a 1 -b ounde d function satisfying Pr I ⊆ ε [ n ] ˜ x ∼ ν ¯ I  ∃ P : Σ I → C a pr o duct function with ∥ P ∥ 2 ≤ 1 such that |⟨ f ¯ I → ˜ x , P ⟩| ≥ ε  ≥ ε, then ther e exist a 1 -b ounde d pr o duct function P : Σ n → C and L : (Σ n , ν ⊗ n ) → C with ∥ L ∥ 2 = 1 and deg ( L ) ≤ d such that |⟨ f , L · P ⟩| ≥ δ . Com bining Theorems 3.8 and 3.9 answers Question 2.6 in the case k = 3 : Theorem 3.10. F or al l ε, α > 0 ther e exist d ∈ N and δ > 0 , such that the fol lowing holds for a p airwise- c onne cte d distribution µ over Σ 3 in which the pr ob ability of e ach atom is at le ast α . If f , g , h : Σ n → C ar e 1 -b ounde d functions such that   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then ther e exist a 1 -b ounde d pr o duct function P : Σ n → C and a function L : Σ n → C with ∥ L ∥ 2 = 1 and deg ( L ) ≤ d such that |⟨ f , L · P ⟩| ≥ δ . 3.3 The Case of k -ary Distributions with No Ab elian Em beddings. Theorem 3.10 turns out to b e suﬃciently strong to complete the argument outlined in Subsection 3.1 for the case that k = 4 and µ has no Ab elian em b eddings [9]. In fact the pap er [9] sho ws that it is suﬃciently strong to address all constan t arities k : Theorem 3.11. Hyp othesis 2.4 is true, and so the answer to Question 2.1 is p ositive if and only if µ admits no Ab elian emb e ddings. 3.4 The Case of k -ary Distributions with Ab elian Em b eddings? At present time, Question 2.6 is op en for all k ≥ 4 , and we b eliev e it is a v ery challenging and interesting problem. Theorem 3.10 suggests a natural and plausible answer for all k , and to state it w e deﬁne the notion of com binatorial low-degree functions: Definition 3.12. W e say a function P : Σ n → C is a c ombinatorial de gr e e- k function if for e ach T ∈  [ n ] k  ther e is a function P T : Σ T → C such that P ( x ) = Q T ∈ ( [ n ] k ) P T ( x T ) for al l x ∈ Σ n , wher e x T ∈ Σ T is the ve ctor whose entries ar e the c o or dinates of x fr om T . Conje ctur e 3.13. F or all k ∈ N there exists k ′ ∈ N , such that for all ε, α > 0 there are d ∈ N and δ > 0 suc h that the following holds. Suppose µ is a pairwise-connected distribution ov er Σ k in which the mass of each atom is at least α . If f 1 , . . . , f k : Σ n are 1 -b ounded functions such that   E ( x 1 ,...,x k ) ∼ µ ⊗ n [ f 1 ( x 1 ) · · · f k ( x k )]   ≥ ε , then there is a combinatorial degree- k ′ function P : Σ n → C with 2 -norm equal to 1 , and L : Σ n → C with 2 -norm equal to 1 and deg ( L ) ≤ d such that |⟨ f 1 , L · P ⟩| ≥ δ . It w ould b e v ery interesting to resolve Conjecture 3.13 either in the p ositiv e or in the negative. If Conjecture 3.13 is true, w e believe that its proof will require developing combinatorial analogs of to ols from additive combinatorics, whic h will b e of indep enden t interest. Subsequently , it will b e in teresting to dev elop density increment argumen ts that w ork with combinatorial low-degree functions, and use it to deduce results that are b ey ond the scop e of the uniformit y norms (we give a few examples in Section 5). If Conjecture 3.13 is false it would b e interesting to examine counter-examples and mo dify Conjecture 3.13 to a plausible statemen t that is still useful. 4 In v erse Theorems for k -Wise Correlations: Pro ofs. In this section we outline some of the ideas in the pro ofs of the statements presented in Section 3. 4.1 The Case of 3 -ary Distributions with No Ab elian Embeddings. 4.1.1 The Reduction to Non-Ab elian Groups: Pro of of Theorem 3.1 The pro of of Theorem 3.1 pro ceeds via a reduction to a problem in non-Abelian F ourier analysis. T o explain the argumen t it will be con venien t to hav e a diﬀerent notation for the alphabet of each input, and we denote the alphab ets of the ﬁrst, second and third inputs by Σ , Γ and Φ resp ectively . W e now explain how to (separately) asso ciate each element in Σ , Γ , Φ with an element in S Σ , the symmetric group ov er Σ . F or Σ , the asso ciation is given b y x → M x when we view the matching M x as a p erm utation. F or Γ and Φ this asso ciation is done via propagation. T ak e y ⋆ ∈ Γ arbitrarily and iden tify it an arbitrary c hosen p erm utation π y ⋆ ∈ S Σ . Next, insp ect the graph G 2 , 3 asso ciated with µ , and propagate the asso ciation via adjacency . Namely , for eac h x we asso ciate the neigh bor z of y ⋆ giv en as z = M x ( y ⋆ ) ∈ Φ with π z = M x ◦ π y ⋆ . Next, for each z already asso ciated with a p ermutation, we consider the neighbor y = M − 1 x ( z ) ∈ Γ for eac h x ∈ Σ and asso ciate it with the p ermutation π y = M − 1 x ◦ π z . W e rep eat this pro cess for suﬃcien tly many steps. 1 It can b e argued that since G 2 , 3 is connected, w e will end up asso ciating all alphab et symbols with p erm utations, so our distribution µ naturally corresp onds to some 3 -ary distribution ov er the p erm utation group S Σ . With some work one can relate the exp ectation (1.2) with a similar exp ectation ov er S n Σ , which can then b e analyzed using elementary non-Ab elian F ourier analysis. 1 During this pro cess it may b e the case that a symbol is asso ciated with tw o distinct p erm utations. Such cases are handled b y operations called “merges”, and ultimately b oil do wn to identifying tw o symbols of Σ , Γ or Φ . 4.1.2 The Inductive Approach: Pro of of Theorem 3.2. The high lev el approac h in the proof of Theorem 3.2 is inspired b y classical tensorization arguments as in [53]. T o start, note that the assumption that µ has no Ab elian embeddings can b e equiv alen tly stated as asserting that there is a constant λ = λ ( µ ) > 0 such that for all 1 -b ounded functions u : Σ → C , v : Γ → C and w : Φ → C with exp ectation 0 , (4.1)   E ( x,y ,z ) ∼ µ [ u ( x ) v ( y ) w ( z )]   ≤ 1 − λ. Indeed, the upper b ound of 1 is trivial by the triangle inequality , and it is tight if and only if for all ( x, y , z ) ∈ supp ( µ ) we hav e that u ( x ) v ( y ) w ( z ) = θ , where θ ∈ C is some constant. Suc h an identit y can b e view ed as an Ab elian em bedding of µ to ([0 , 2 π ) , + (mo d 2 π )) , hence it does not exist if µ has no Abelian em b eddings. With this in mind, it is natural to exp ect that if f , g , h are functions of degree at least large d as in Hyp othesis 2.4, then the univ ariate inequality ab o v e would tensorize and giv e us a b ound of (1 − λ ) d → d →∞ 0 . There are several issues with this approach, though: 1. A base case inequality such as (4.1) do es not tensorize well. Speciﬁcally , the 1 -b oundedness assumption is not well-preserv ed under sp ectral/F ourier-analytic techniques. Instead, one needs a base case for functions b ounded in L 2 -norm. 2. T ensorizing inequalities such as (4.1), namely inequalities inv olving a pro duct of 3 functions, is far from b eing automatic. This is in contrast to similar inequalities inv olving only tw o functions u, v , whic h can b e stated in terms of eigenv alues of some matrix and thus tensorizes well (as w as done by Mossel [53]). T o o vercome these issues, the work [15] ﬁrst establishes an alternative base case for univ ariate functions b ounded in L 2 -norm. Due to its technical nature we refer the reader to [15] for a precise statemen t of this base case, but remark that it is not simply the assertion that (4.1) holds for functions with L 2 -norm at most 1 (w e do not know how to deduce such a base case from the assumption that µ has no Ab elian embeddings). The base case of [15] is in fact a statement ab out a distribution µ ′ diﬀeren t from µ , for whic h exp ectations of the form (1.2) o ver µ can b e upp er b ounded by similar exp ectations ov er µ ′ . The main argument of [15] pro ceeds by induction on n using the singular-v alue decomp osition (SVD in short), asserting that f : Σ n → C , g : Γ n → C and h : Φ n → C with 2 -norm equal to 1 can b e written as f ( x ) = X i λ i u i ( x 1 ) f i ( x 2 , . . . , x n ) , g ( y ) = X i γ i v i ( y 1 ) g i ( y 2 , . . . , y n ) , h ( z ) = X i κ i w i ( z 1 ) h i ( z 2 , . . . , z n ) , where each one of { u i } , { f i } , { v i } , { g i } , { w i } , { h i } i is an orthonormal system and λ i , γ i , κ i are non-negative real n umbers satisfying P i λ 2 i = P i γ 2 i = P i κ 2 i = 1 . If eac h one of the SVDs inv olv es only a single summand, then E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )] = E ( x 1 ,y 1 ,z 1 ) ∼ µ [ u 1 ( x ) v 1 ( y ) w 1 ( z )] E ( x,y ,z ) ∼ µ ⊗ ( n − 1) [ f 1 ( x ) g 1 ( y ) h 1 ( z )] . T o complete the inductiv e step the ﬁrst exp ectation can b e bounded using the base case, and the second exp ectation can b e b ounded using the inductive hypothesis. If the SVDs of f , g and h inv olv e more than a single term, then there are many additional terms that need to b e upp er b ounded, and the argument pro ceeds by considering tw o sub cases. If there are SVDs so that essentially all of the mass of f , g and h lies on a single component, then one sho ws that the other terms do not interfere m uch, and one essentially only has to account for the contribution of the main term as ab ov e. Otherwise, it can b e sho wn that the degrees of f , g and h all must b e Θ( n ) , and one pro ceeds via a diﬀeren t inductive argumen t. One carefully c ho oses F , G, H whic h are random linear combinations of f i , g i , h i resp ectiv ely (these are ( n − 1) -v ariate functions), and shows that the 3 -wise correlation of f , g , h according to µ ⊗ n is at most (1 − Ω(1)) times the 3 -wise correlation of F , G, H according to µ ⊗ ( n − 1) . As f , g , h are Θ( n ) degree F, G, H are also Θ( n ) degree, so one can induct Ω( n ) times, completing the pro of. 4.2 The Case of No ( Z , +) -embeddings: Pro of of Theorem 3.4. By considering the set of equations deﬁning Ab elian em b eddings as a linear system of equations, one can show that if µ admits no ( Z , +) -em b eddings, then all of its embeddings are equiv alen t to an embedding into G = Q p ≤ p max ,ℓ ≤ ℓ max Z p ℓ , where p max , ℓ max dep end only on | Σ | . This will be the group G in the theorem, and one ﬁxes a canonical embedding ( σ , γ , ϕ ) of µ in to G . By that, we mean an embedding that encapsulates within it all of the Ab elian embeddings of µ . The pro of of Theorem 3.4 follows the spirit of Theorem 3.2, but there are man y diﬀe rences: 1. The base case inequality along the lines of (4.1) cannot be true if µ has any Ab elian embedding (indeed, applying a character on any Ab elian em b eddings gives a coun ter example). Instead, one deﬁnes the space of univ ariate “em b edding functions”, which are functions of the form χ ◦ σ , χ ◦ γ , χ ◦ ϕ for χ ∈ ˆ G , and requires a base case inequality along the lines of (4.1) when at least one of the functions u, v , w is far from b eing an em b edding function. 2. In the inductiv e arguments, one considers a more reﬁned degree notion for f , g , h , called the non-emb e dding de gr e e . Roughly sp eaking, one deﬁnes a basis for L 2 (Σ , µ 1 ) (and similarly for L 2 (Σ , µ 2 ) and L 2 (Σ , µ 3 ) ) consisting of (1) em b edding functions, (2) functions orthogonal to all embedding functions, and then tensorizes it to get a basis for L 2 (Σ n , µ ⊗ n 1 ) . One can then write f 1 (and similarly f 2 , f 3 ) according to this basis, and consider separately the notions of (1) em b edding degree (which is the n umber of functions from the ﬁrst part of the basis), (2) the non-em b edding degree (which is the num ber of functions from the second part of the basis). The proof of Theorem 3.4 pro ceeds via similar SVD-based inductive arguments using the notion of non-em b edding degree instead of the standard notion of degree ab o ve. The conclusion from the inductiv e arguments is that the non-em b edding degree of each one of f , g , h m ust b e small. Th us, after a random restriction their non-embedding degree b ecomes constant, meaning they are essentially functions ov er the group G ! The 3 -wise correlation of the restrictions can now b e studied using F ourier analysis ov er the group G , leading to the conclusion of Theorem 3.4. 4.3 The P airwise-connected Case: Proof of Theorem 3.8. While the pro of of Theorem 3.8 uses some ideas from the argumen ts discussed ab o ve, the ov erarc hing pro of strategy is completely diﬀerent. The main idea is to identify a norm, called the swap norm , and show that: 1. it (morally) dominates all 3 -wise correlations with resp ect to a pairwise-connected distribution µ , and 2. it admits an inv erse theorem in the spirit of Theorem 3.8. The deﬁnition of this norm is indep endent of µ , meaning it simultaneously w orks for pairwise-connected 3 -ary distribution. This is reminiscent of the notion of “Cauc h y-Sch w arz” complexit y [32], asserting that the Gow ers uniformit y norms dominate correlations deﬁned by collections of linear forms. 4.3.1 The Sw ap Norm. T o deﬁne the swap norm we require the deﬁnition of the b ox-norm. Belo w, the notation x ∈ R Σ n means that x is sampled uniformly from Σ n , and the notation I ⊆ 1 / 2 [ n ] means that eac h co ordinate i ∈ [ n ] is included in I with probability 1 / 2 . Definition 4.1. L et f 1 , f 2 , f 3 , f 4 : Σ n → C b e functions, and let I ⊆ [ n ] b e a set of c o or dinates. The b ox form b o x I ( f 1 , f 2 , f 3 , f 4 ) is deﬁne d as b o x I ( f 1 , f 2 , f 3 , f 4 ) = E x,x ′ ∈ R Σ I y ,y ′ ∈ R Σ ¯ I [ f 1 ( x, y ) f 2 ( x ′ , y ′ ) f 3 ( x, y ′ ) f 4 ( x ′ , y )] . The b ox norm of a function f : Σ n → C with r esp e ct to I ⊆ [ n ] is deﬁne d as box I ( f ) = b o x ( f , f , f , f ) 1 / 4 . Definition 4.2. L et f 1 , f 2 , f 3 , f 4 : Σ n → C b e functions. The swap form swap ( f 1 , f 2 , f 3 , f 4 ) is deﬁne d as sw ap ( f 1 , f 2 , f 3 , f 4 ) = E I ⊆ 1 / 2 [ n ] [ b o x I ( f 1 , f 2 , f 3 , f 4 )] . The swap norm of a function f : Σ n → C is deﬁne d as sw ap ( f ) = sw ap ( f , . . . , f ) 1 / 4 . The b o x form is w ell known to hav e many useful properties suc h as the Cauch y-Sch w arz-Go wers inequality [31]. Being the av erage of b o x forms, the swap form also satisﬁes similar prop erties, suc h as: | sw ap ( f 1 , f 2 , f 3 , f 4 ) | 2 ≤ swap ( f 1 , f 2 , f 1 , f 2 ) sw ap ( f 3 , f 4 , f 3 , f 4 ) , | sw ap ( f 1 , f 2 , f 1 , f 2 ) | 2 ≤ swap ( f 1 ) sw ap ( f 2 ) . These prop erties can b e used to show that the swap norm is in fact a norm. The source of the name “swap norm” comes from the following identit y . F or x, y ∈ Σ n , let x ← → y be the distribution o ver ( u, v ) ∈ Σ n × Σ n where for eac h i ∈ [ n ] independently , with probabilit y 1 / 2 we hav e ( u i , v i ) = ( x i , y i ) , and else w e hav e ( u i , v i ) = ( y i , x i ) . With this notation, it can b e easily veriﬁed by expanding the deﬁnition that sw ap ( f 1 , f 2 , f 3 , f 4 ) = E x,y ∈ R Σ n ( u,v ) ∼ ( x ↔ y ) h f 1 ( x ) f 2 ( y ) f 3 ( u ) f 4 ( v ) i . The connection betw een 3 -wise correlations and the sw ap norm comes from the following lemma, asserting that (under random restrictions) the sw ap norm of a function f dominates 3 -wise correlations: Lemma 4.3. L et µ b e a p airwise-c onne cte d distribution over Σ 3 in which the pr ob ability of e ach atom is at le ast α , and write µ 1 = αU + (1 − α ) ν , wher e µ 1 is the mar ginal distribution of µ on its ﬁrst c o or dinate, U is the uniform distribution over Σ and ν is some distribution over Σ . Then ther e exists C = C ( α ) > 0 such that for al l 1 -b ounde d functions f , g , h : Σ n → C it holds that   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   C ( α ) ≤ E I ⊆ α [ n ] w ∼ ν ¯ I  sw ap ( f ¯ I → w ) 4  . The pro of of Lemma 4.3 is by a symmetry argument inv olving several applications of the Cauch y-Sc h warz inequalit y (as well as random restrictions), and w e do not give it here. 4.3.2 An In verse Theorem for the Swap Norm. By an av eraging argument, Lemma 4.3 asserts that if   E ( x,y ,z ) ∼ µ ⊗ n [ f ( x ) g ( y ) h ( z )]   ≥ ε , then with noticeable probability ov er the c hoice of I and w we hav e that sw ap ( f ¯ I → w ) is noticeable. Thus, Theorem 3.8 b oils down to the following inverse the or em for the swap norm : Theorem 4.4. F or al l ε > 0 ther e exists δ > 0 such that the fol lowing holds. Supp ose that f : Σ n → C is a 1 -b ounde d function such that swap ( f ) ≥ ε . Then Pr I ⊆ δ [ n ] ˜ x ∈ R Σ ¯ I  ∃ P : Σ I → C a pr o duct function with ∥ P ∥ 2 ≤ 1 such that |⟨ f ¯ I → ˜ x , P ⟩| ≥ δ  ≥ δ. T o get some intuition for Theorem 4.4, it is useful to think ab out the b ox norm and the following (rather elemen tary) inv erse theorem for it. If b o x I ( f ) ≥ ε , then b y deﬁnition w e may ﬁnd x ′ ∈ Σ I and y ′ ∈ Σ ¯ I suc h that        f ( x ′ , y ′ ) E x ∈ R Σ I y ∈ R Σ ¯ I [ f ( x, y ) f ( x, y ′ ) f ( x ′ , y )]        ≥ b o x I ( f ) ≥ ε. Using the 1 -boundedness of f and deﬁning the functions g I : Σ I → C and h ¯ I : Σ ¯ I → C by g ( x ) = f ( x, y ′ ) , h ( y ) = f ( x ′ , y ) , w e conclude that |⟨ f , g I h ¯ I ⟩| ≥ ε . In words, if f has a large b ox norm with resp ect to I ⊆ [ n ] , then it is correlated with a pro duct of 2 functions, one dep ending only on the coordinates of I and the other depending only on the co ordinates of ¯ I . Returning to the swap norm, an a veraging argument gives that if swap ( f ) ≥ ε , then for I ⊆ 1 / 2 [ n ] with noticeable probability it holds that b ox I ( f ) ≥ ε/ 2 . Applying the in v erse theorem for the b ox norm, we get that for I ⊆ 1 / 2 [ n ] , with noticeable probability the function f is correlated with a function of the form g I h ¯ I . It is easy to see that pro duct functions as in Deﬁnition 3.7 are natural examples of functions f satisfying this prop ert y . A closer insp ection sho ws that functions of essentially low-degree may also satisfy this prop ert y (for example, the ma jority function). This means that deducing the pro duct-function structure cannot b e straightforw ard and still requires some eﬀort. In particular, the random-restriction part in Theorem 4.4 is still essential. A natural attempt at proving Theorem 4.4 pro ceeds via an energy-increment argument using the b o x norm in verse result. F rom the ab o ve argumen t, it is natural to try to prov e that if swap ( f ) ≥ ε , then there is a set I ⊆ [ n ] and functions g 1 , h 1 as ab o ve such that swap ( f − λ 1 g 1 h 1 ) is noticeably smaller than swap ( f ) . Contin uing in this fashion, one can hop e to ﬁnd sets I i ⊆ [ n ] , functions ( g i , h i ) and λ i ∈ (0 , 1] such that sw ap ( f − T P i =1 λ i g i h i ) is small, say smaller than ε/ 10 , and T = O ε (1) . In that case the triangle inequalit y implies sw ap ( T P i =1 λ i g i h i ) ≥ 9 ε/ 10 . It is useful to consider the simple case that T = 1 . In this case, the last inequality means that sw ap ( g 1 ) sw ap ( h 1 ) = swap ( g 1 h 1 ) ≥ 9 ε/ 10 , so either sw ap ( g 1 ) or swap ( h 1 ) is signiﬁcantly bigger than ε . Since b oth functions can b e view ed as random restrictions of f , we conclude that a suitably c hosen random restriction of f has swap-norm noticeably bigger than that of f . This facilitates an inductive approach. In the case that T > 1 the argument is not as straight-forw ard. Instead of showing that there is i such that either g i or h i ha ve noticeably bigger swap norms, one shows that there is complex combination g ′ = T P i =1 θ i g i or h ′ = T P i =1 θ i h i with T P i =1 | θ i | 2 = 1 that achiev es swap-norm noticeably bigger than ε . A dditionally , one shows that if either g ′ or h ′ is correlated with a pro duct function, then a random restriction of f outside I 1 ∩ . . . ∩ I T is also correlated with a pro duct function. T ogether, these tw o facts facilitate an inductive approac h as well. W e remark that the formal pro of in [8] inv olv es several signiﬁcant technical c hallenges: 1. First, we do not actually kno w how to sho w that the swap norm of f − g 1 h 1 is noticeably smaller than the swap norm of f itself, so it is not clear how to argue that the pro cess will terminate within T = O ε (1) steps based on this consideration alone. T o address this issue one incorporates an L 2 -norm based energy incremen t approach, which is useful since ∥ f − g 1 h 1 ∥ 2 2 is indeed noticeably smaller than ∥ f ∥ 2 2 . 2 2. Second, in the case T > 1 obtaining a relation b etw een f and pro duct functions and g ′ , h ′ is not clear for arbitrary functions g 1 , . . . , g T and h 1 , . . . , h T . Instead, one uses SVDs of f to ﬁnd the comp onen ts g i , h i . The upshot here is that taking ( g 1 , h 1 ) to b e SVD components of f with resp ect to the partition [ n ] = I 1 ∪ ¯ I 1 , one has g 1 ( x ) = E y ∈ R Σ ¯ I 1 [ f ( x, y ) h 1 ( y )] , so the correlation of g 1 with a pro duct function P can b e related to the correlation of f ¯ I 1 → y with P for a random y ∈ R Σ ¯ I 1 . 3. Third, the fact that the sets I i ⊆ [ n ] are distinct complicates the ab ov e argument considerably , as the functions g i h i are not compatible with each other and it is not clear how to relate them to a common SVD of f . T o address this issue one applies a random restriction to “pro ject” the sets I i in to a common partition ( S, T ) , so that they (morally) all b ecome SVD comp onents with resp ect to the same partition. 5 Applications. In this section we discuss some applications of the results from Section 3. 5.1 A dditiv e Combinatorics. W e b egin with applications in additive combinatorics, and more s peciﬁcally to the study of restricted v ersions of the arithmetic progression patterns from Section 1. 5.1.1 Restricted Arithmetic Progressions of Length 3 . Definition 5.1. L et p ≥ 3 b e a prime and let n ∈ N b e a lar ge inte ger. 1. W e say that a triplet x, x + a, x + 2 a forms a somewhat-r estricte d 3 -AP if x ∈ F n p and a ∈ { 0 , 1 , 2 } n \ {  0 } . W e say that A ⊆ F n p is somewhat-r estricte d 3 -AP fr e e if A c ontains no somewhat-r estricte d 3 -AP. 2. W e say that a triplet x, x + a, x + 2 a forms a r estricte d 3 -AP if x ∈ F n p and a ∈ { 0 , 1 } n \ {  0 } . W e say that A ⊆ F n p is r estricte d 3 -AP fr e e if A c ontains no r estricte d 3 -AP. It is clear that for an y set A ⊆ F n p w e hav e that A is 3 -AP free = ⇒ A is somewhat-restricted 3 -AP free = ⇒ A is restricted 3 -AP free . It is thus natural to ask: What is the largest density of a somewhat-restricted 3 -AP free set? What is the largest densit y of a restricted 3 -AP free set? It also makes sense to consider the counting versions of these problems: if A has densit y α > 0 (thought of as a constant), must it con tain Ω α (1) fraction of the somewhat-restricted 3 -APs? Must it contain Ω α (1) fraction of the restricted 3 -APs? One motiv ation for studying these problems is that the F ourier-analytic approac h, as outlined in Section 1, no longer w orks, so making progress on these problems is likely to require a diﬀeren t approach. The density Hales- Jew ett theorem [28, 29, 22, 24] implies that a set A ⊆ F n p with no restricted 3 -APs has v anishing density o (1) . 2 Combining the swap-norm and the L 2 -norm energy increment strategies amount to deﬁning a certain p otential function that takes both into account, and measuring how it changes throughout the pro cess. The quan titative b ounds it giv es though are either ineﬀective (if one uses the ergo dic-theoretic pro of of [28, 29]), or else v ery weak inv erse-to wer type (if one uses the combinatorial pro of of [22, 24]). The counting versions, for whic h the density Hales-Jewett do es not apply , hav e b een raised in [40] and in [35] and remain completely op en. In [12] the authors combine Theorem 3.6 with an appropriate densit y increment argument and sho w: Theorem 5.2. A somewhat-r estricte d 3 -AP fr e e set A ⊆ F n p has density at most O  1 (log log log n ) Ω p (1)  . The pro of of Theorem 5.2 proceeds b y ﬁrst translating the assumption that A is somewhat-restricted 3 -AP set to an inequality as (1.1), except that a is sampled uniformly from { 0 , 1 , 2 } n . Using the fact that the uniform distribution ov er { ( x, x + a, x + 2 a ) | x ∈ F p , a ∈ { 0 , 1 , 2 }} has no ( Z , +) -embeddings they apply Theorem 3.6 to conclude that f A = 1 A − µ ( A ) is correlated with a function of the form L · χ ◦ σ ⊗ n . Using random restrictions one can get rid of the lo w-degree term L , and for simplicity we ignore it. As each χ j is it a character ov er a group G of size O p (1) there are only W = O p (1) many options for eac h χ j . Therefore, w e may partition the co ordinates j ∈ [ n ] into groups J 1 , . . . , J W dep ending on χ j . The k ey observ ation is no w that b y taking | G | co ordinates in some J i and “identifying them”, namely plugging the same v alue of v ∈ F p in all of them, the function χ ◦ σ no longer dep ends on them (as together they contribute ( χ ◦ σ ( v )) | G | = 1 , where χ is the character applied on co ordinates from J i ). Using this idea one can decomp ose F n p in to copies of F n ′ p on whic h χ ◦ σ is constan t and n ′ ≥ Ω p ( n ) , giving a density increment for A on one of these copies. The case of restricted 3 -APs is more diﬃcult, since the uniform distribution ov er { ( x, x + a, x + 2 a ) | x ∈ F p , a ∈ { 0 , 1 }} do es contain ( Z , +) -embeddings. Still, it is pairwise-connected, and in particular one can apply Theorem 3.8 or Theorem 3.10. Indeed, using these results along with a similar-in-spirit densit y increment argumen t, Theorem 5.2 is strengthened in [8] to the follo wing result: 3 Theorem 5.3. A r estricte d 3 -AP fr e e set A ⊆ F n p has density at most O  1 (log log log n ) Ω p (1)  . 5.1.2 Com binatorial Lines of Length 3 . There is a v arian t of the 3 -APs pattern that is ev en more restrictiv e than restricted 3 -APs. Due to the lack of arithmetic structure, it is more often referred to as c ombinatorial lines . Definition 5.4. L et k ≥ 3 b e an inte ger. A c ombinatorial line of length k in { 0 , 1 . . . , k − 1 } n is a k -tuple x 1 , . . . , x k of distinct p oints in { 0 , 1 . . . , k − 1 } n such that for e ach c o or dinate j ∈ [ n ] ( x 1 ( j ) , . . . , x k ( j )) ∈ { (0 , . . . , 0) , (1 , . . . , 1) , . . . , ( k − 1 , . . . , k − 1) , (0 , 1 , 2 , . . . , k − 1) } . In w ords, a combinatorial line of length k is a k -tuple x 1 , . . . , x k of distinct p oin ts such that on each co ordinate j , either all entries are equal, or else they must satisfy x 1 ( j ) = 0 , x 2 ( j ) = 1 , . . . , x k ( j ) = k − 1 . The problem of determining the density of the largest A ⊆ { 0 , 1 , . . . , k − 1 } n with no combinatorial lines of length k is known in the literature as the density Hales-Jewett problem, or DHJ- k in short (b eing the density v ersion of the classical Hales-Jewett theorem [38]). It has received signiﬁcant attention ov er the years for several reasons. First, it is arguably the most strict combinatorial structure one may hop e to ﬁnd inside any dense set A . Second, one can sho w that a set of integers A ⊆ { 1 , . . . , n } with no k -APs can b e translated into a set of p oints A ′ ⊆ { 0 , 1 . . . , k − 1 } n ′ with similar density and no com binatorial lines of length k . In particular, showing that a set with no combinatorial lines of length k m ust hav e v anishing density implies Szemerédi’s theorem. Using ergo dic-theoretic techniques, F urstenberg and Katznelson show ed that a set A ⊆ { 0 , 1 . . . , k − 1 } n with no combinatorial lines of length k has v anishing density . The ﬁrst work to establish quantitativ e b ounds was done by the Polymath1 pro ject [22], who established inv erse-tow er t yp e density b ounds. F or the case k = 3 , their b ound was of the order 1 / p log ∗ n , where log ∗ n is the num b er of times that log needs to b e applied on n until one gets b elow 1 . In [10] the authors establish the ﬁrst known “reasonable b ound” for the DHJ- 3 problem, meaning a b ound with ﬁnitely many logarithms: Theorem 5.5. A set A ⊆ { 0 , 1 , 2 } n with no c ombinatorial lines of length 3 has density at most O  1 (log log log log n ) Ω(1)  . The pro of of Theorem 5.5 uses Theorem 3.8 in non-ob vious w a y; in fact, the pro of also uses Theorem 3.11. The main issue is that distributions µ relev an t to the study of combinatorial lines are only supp orted 3 W e remark that similar results hold for a more general family of patterns of length 3 . on { (0 , 0 , 0) , (1 , 1 , 1) , (2 , 2 , 2) , (0 , 1 , 2) } , so they are not pairwise-connected. Therefore, one cannot (directly) apply Theorem 3.8 to get any structural result on f A = 1 A − | A | / 3 n if A has no combinatorial lines of length 3 . T o remedy this, the authors use ideas from Shkredov’s argumen t for the corners problem [60] (see also [34, 33]). In the corners problem one w an ts to upp er b ound the density of a set A ⊆ F n p × F n p that contains no pattern of the form ( x, y ) , ( x + a, y ) , ( x, y + a ) . F or the integer version of the problem, a direct density increment approach w as used by [1], who show ed that the density of a corner-free set is o (1) . The quantitativ e b ound was signiﬁcantly impro ved by Shkredov’s [60] via a r elative density increment argument. His idea w as that, instead of measuring the density of A in comparison to the ambien t space, one should measure it with resp ect to a rectangle E 1 × E 2 where E 1 , E 2 ⊆ F n p . The upshot of this approac h is that it is relativ ely straigh tforward to show that if A ⊆ F n p × F n p has no corners, then it admits a density incremen t on some E 1 × E 2 for sizable E 1 , E 2 . The downside is that it is tric ky to iterate this sort of density increment statement. T o do so, Shkredov ensures that E 1 , E 2 satisfy suitable pseudo-randomness prop erties, which then allows him to iterate the argument (see [41] for a recent impro v ement). The pro of of Theorem 5.5 follows the high-level approach of Shkredo v’s argument. The argumen t uses a “dictionary” betw een notions in the context of corners (such as rectangles) to notions in the context of DHJ (such as 0 , 1 and 0 , 2 -insensitive sets and the disjoint pro duct), that was in fact already observ ed by [22]. The main high-lev el diﬀerence b et w een the argument of [10] and of [60] is in the pseudo-random notion that is used. In the con text of corners, the correct notion turns out to be having small U 2 uniformit y norm, whereas in the con text of DHJ- 3 , the correct notion turns out to (morally) b e having a small swap norm. 4 5.2 Complexit y Theory . W e next discuss some applications to problems in complexity theory . 5.2.1 The Hybrid Algorithm for Satisﬁable CSPs. In Theorem 1.5, Raghav endra shows a dichotom y result for approximating CSPs with a small loss in the completeness parameter. As explained therein, a k ey comp onen t in his pro of is a dictatorship test with matching parameters. In [9] the authors sho w that Ragha vendra’s dictatorship test construction remains v alid with no loss in the completeness parameter for predicates P : Σ k → { 0 , 1 } such that P − 1 (1) has no Ab elian embeddings: Theorem 5.6. Supp ose that P is a c ol le ction of k -ary pr e dic ates such that P − 1 (1) has no Ab elian emb e ddings for al l P ∈ P . 5 Then ther e exists s ∈ [0 , 1] such that: 1. A lgorithm: ther e is a p olynomial time algorithm that solves gap- P -CSP [1 , s ] . 2. Evidenc e for Har dness: ther e is a dictatorship test for P with c ompleteness 1 and soundness s + o (1) . The algorithm in Theorem 5.6 is the same as in Theorem 1.5 and is based on semi-deﬁnite programming. F or predicates that ha ve Ab elian em b eddings, it is well known that the optimal approximation algorithm cannot b e based only on semi-deﬁnite programming relaxations [37, 59]. As of the time of writing, there is no clear candidate approximation algorithm in this case, and many sp ecial cases are still op en. In [16] the authors cons ider 3 -ary predicates P that admit Abelian embeddings but no ( Z , +) -em b eddings. F or suc h predicates P the authors propose a p olynomial time “hybrid algorithm”, which is an intert wined com bination of a semi-deﬁnite programming algorithm and a Gaussian elimination algorithm, as a candidate optimal approximation algorithm. Their result reads: Theorem 5.7. Supp ose that a c ol le ction of 3 -ary pr e dic ates P satisﬁes that P − 1 (1) has no ( Z , +) -emb e ddings for al l P ∈ P . 6 Then ther e exists s ∈ [0 , 1] such that: 1. A lgorithm: ther e is a p olynomial time algorithm for gap- P -CSP [1 , s ] . 2. Evidenc e for Har dness: ther e is a dictatorship test for P with c ompleteness 1 and soundness s + o (1) . While w e do not outline the pro of of Theorem 5.7, w e mention that it uses a generalization of the inv ariance principle of [54] called the mixe d invarianc e principle . Whereas the in v ariance principle of [54] relates the b ehavior of low-degree p olynomials b etw een discrete space and Gaussian space, the mixed inv ariance principle relates the 4 Strictly sp eaking, for technical reasons the necessary pseudo-randomness condition is that the function has no correlation with product functions as in Deﬁnition 3.7, even after randomly restricting all but √ n of the co ordinates. 5 Strictly sp eaking, the required condition is more technical and asserts that in an optimal SDP solution, all lo cal distributions hav e no Abelian embeddings. W e refrain from stating it explicitly for the sake of simplicity . 6 Strictly sp eaking, the required condition is more technical and asserts that in an optimal SDP solution, all lo cal distributions hav e no ( Z , +) -embeddings. W e refrain from stating it explicitly for the sake of simplicity . b eha vior of the wider family of “low-degree p olynomials times character function” (as app earing in Theorem 3.6) b et w een discrete space and the pro duct of Gaussian and Ab elian group spaces. 5.2.2 Multipla y er P arallel Rep etition. The parallel repetition theorem of Raz [57] is a p ow erful to ol in theoretical computer science. T o demonstrate it, consider a 2 -pla y er game based on graph coloring: supp ose that G = ( V , E ) is a graph that is either 3 -colorable (meaning there is χ : V → { 0 , 1 , 2 } such that χ ( u )  = χ ( v ) for all ( u, v ) ∈ E ) or is ε -far from b eing 3 -colorable (meaning that for all χ : V → { 0 , 1 , 2 } it holds that χ ( u ) = χ ( v ) for at least ε fraction of the edges ( u, v ) ∈ E ). Consider the graph G ⊗ n whose vertices are V n , and whose edges are (( u 1 , . . . , u n ) , ( v 1 , . . . , v n )) suc h that ∀ i ( u i , v i ) ∈ E . Supp ose w e wish to lab el V n b y { 0 , 1 , 2 } n so that for eac h edge (  u,  v ) , the assignment of the endp oin ts diﬀer on all co ordinates. If G is 3 -colorable, it is easily seen that suc h assignment for G ⊗ n exists. Th e parallel rep etition theorem asserts that if G is ε -far from b eing 3 -colorable, then any assignment to G ⊗ n satisﬁes at most (1 − Ω ε (1)) n fraction of the edges. Pro ving multipla y er parallel rep etition theorems turns out to b e signiﬁcan tly more challenging. Here, a k - pla yer game I consists of a k -partite k -uniform hypergraph G = ( V 1 ∪ . . . ∪ V k , E ) , alphab ets Σ 1 , . . . , Σ k and a constrain t Φ e ⊆ Σ 1 × . . . Σ k for each edge e ∈ E . The v alue of I is deﬁned as val ( I ) = max A i : V i → Σ i |{ e = ( v 1 , . . . , v k ) ∈ E | ( A 1 ( v 1 ) , . . . , A k ( v k )) ∈ Φ e }| | E | . The n -fold repeated game I ⊗ n is given by the k -partite k -uniform hypergraph ( V 1 n ∪ . . . ∪ V k n , E ′ ) with the edge set E ′ = { (  v 1 , . . . ,  v k ) | (  v 1 ( j ) , . . . ,  v k ( j )) ∈ E ∀ j = 1 , . . . , n } , alphabets Σ n 1 , . . . , Σ n k and constrain ts Φ e ′ ⊆ Σ n 1 × . . . × Σ n k for e ′ ∈ E ′ . W e view eac h a label from Σ n 1 to a v ertex  v 1 as assigning a lab el from Σ 1 to eac h one of the co ordinates of  v 1 . In this language, the constraint Φ e ′ on an edge e ′ = (  v 1 , . . . ,  v k ) imposes that for each j = 1 , . . . , n , the lab els giv en to the j th co ordinate of  v 1 , . . . ,  v k satisfy the constraint Φ e j where e j = (  v 1 ( j ) , . . . ,  v k ( j )) ∈ E . With this terminology , [27] suggested to study the multipla y er parallel rep etition conjecture: if I is a k -pla yer game such that val ( I ) ≤ 1 − ε and ε > 0 , then val ( I ⊗ n ) ≤ (1 − ε ′ ) n where ε ′ > 0 dep ends only on k , ε (and p ossibly other parameters of the base game I suc h as the alphab et sizes). In an imp ortan t work Raz [57] sho wed, using information theoretic techniques, that this statement is correct for k = 2 . The case k ≥ 3 remains wide op en to date. The techniques of Raz work for the class of “connected” k -play er games [23], but they fail for any disconnected game (the deﬁnition of a connected game is analogous to the deﬁnition of a connected distribution). Recen t works [18, 7, 6] made some progress on the m ultipla yer parallel rep etition conjecture, and used the analytic machinery discussed in Section 3 to prov e exp onential decay rates for some classes of 3 -play er games. 6 Op en Problems. W e ﬁnish this article by men tioning a few op en problems for future researc h. The ﬁrst problem is to prov e a general inv erse theorem for correlations ov er pairwise-connected distributions: Pr oblem 6.1. R esolve Conje ctur e 3.13 . W e remark that ev en resolving sp ecial cases of Conjecture 3.13 (for k ≥ 4 ) w ould b e signiﬁcan t progress. The next problem is to establish eﬀectiv e b ounds for the density Hales-Jewett problem for k ≥ 4 : Pr oblem 6.2. Show that if A ⊆ { 0 , 1 . . . , k − 1 } n has no c ombinatorial lines, then the density of A is at most O  1 log ... log n  wher e the numb er of applic ations of log is O k (1) . W e exp ect that to resolve Problem 6.2 one would (at the very least) need to resolve Problem 6.1, but susp ect more work to b e needed. W e b eliev e it w ould b e already interesting to ﬁnd a density increment strategy (or any other strategy) that resolves Problem 6.2 assuming a statemen t along the lines of Conjecture 3.13. Man y developmen ts in the study of the Gow ers uniformity norms hav e b een motiv ated by ergo dic theory , and we b elieve that there may b e an in teresting ergo dic theoretic view of the general correlations studied in this article. It would b e interesting to ﬁnd such a connection and perhaps use it to prov e inv erse ty p e results in the spirit of Theorems 3.4 and 3.8 (even with no quantitativ e b ounds): Pr oblem 6.3. Find an er go dic-the or etic appr o ach for the study of k -wise c orr elations as in (1.2) and for establishing inverse the or ems as discusse d in this article. F rom the p ersp ectiv e of theoretical computer science, p erhaps the most am bitious goal with regards to the topics in this article would b e to establish a dichotom y result for approximating satisﬁable CSPs: Pr oblem 6.4. Show that the fol lowing dichotomy r esult: for al l k ∈ N , ﬁnite alphab et Σ and P c ol le ction of k -ary pr e dic ates over Σ such that P -CSP is NP-har d, ther e is an s ∈ (0 , 1) such that: 1. A lgorithm: ther e is a p olynomial time algorithm for gap- P -CSP [1 , s ] . 2. Har dness: for al l δ , the pr oblem gap- P -CSP [1 , s + δ ] is NP-har d, p ossibly assuming a c onje ctur e such as the R ich 2 -to- 1 Games Conje ctur e [17]. A bit less ambitiously, show a dictatorship test for P with c ompleteness 1 and soundness s + δ . Next, it would b e interesting to ﬁnd an analytical proof of the dichotom y theorem of Zh uk and Bulato v [66, 19]. Besides b eing a diﬀeren t pro of it may b e the case that an analytical approach would giv e a diﬀerent (possibly simpler) algorithm: Pr oblem 6.5. Give an analytic al pr o of for the dichotomy the or em: for al l k ∈ N , ﬁnite alphab et Σ and a c ol le ction of k -ary pr e dic ates P over Σ , the pr oblem P -CSP either admits a p olynomial time algorithm or else is NP-har d. W e expect b oth Problem 6.4 and Problem 6.5 to be very c hallenging, and think that analyzing sp eciﬁc classes of predicates would already b e very in teresting. In particular, at present it is not clear what an algorithm as in Problem 6.4 should b e. The next problem is to make further progress on the multipla y er parallel repetition conjecture. The results discussed in Subsection 5.2.2 apply only to the class of X OR games, and it is not clear ho w to use an y of these analytical techniques for other classes of multipla y er games (ev en for a small num b er of play ers, say 3 ): Pr oblem 6.6. Show that if I is a k -player game with val ( I ) ≤ 1 − ε , then val ( I ⊗ n ) ≤ (1 − ε ′ ) n wher e ε ′ = ε ′ ( ε, I , k ) > 0 . The b est known result in the direction of Problem 6.6 is by [65], who show ed that lim n →∞ val ( I ⊗ n ) = 0 , with no explicit quantitativ e b ounds. Establishing any reasonable bounds in Problem 6.6 would already b e signiﬁcant. T ensorization in diﬀer ent c ontexts? It is a very common phenomenon that tensorizing inequalities that are not captured by eigenv alues of matrices is very challenging. In fact, this is partly the reason problems in multipla yer comm unication complexity , and regarding explicit constructions of high-rank tensors, are so challenging. W e b eliev e it w ould b e v ery interesting to see if there are any approaches along the lines describ ed in Section 4 that ma y b e applicable in these contexts (or others). 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