Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

Rank Bounds for Design Matrices with Applications t o Com binatorial Geometr y and Lo cally Corr ectable Co des Boaz Barak ∗ Zeev Dvir † Avi Wigderson ‡ Amir Y eh uda y oﬀ § Abstract A ( q , k , t )-design matrix is an m × n matrix whose pattern of zeros/non-zer os satisﬁes the following design-like condition: eac h row has at most q non-zer os, each column has at le a st k non-zeros and the supp orts o f every tw o columns intersect in at most t r ows. W e prov e that for m ≥ n , the rank of any ( q , k , t )-desig n matrix over a ﬁeld of characteristic zero (or suﬃciently large ﬁnite characteristic) is at least n −  q tn 2 k  2 . Using this result we der ive the following a pplications: Imp oss ibili ty resul ts for 2 -query LCCs ov er large ﬁelds. A 2-query lo cally corr e ctable co de (LCC) is an er ror correcting co de in which ev ery co deword co ordinate can be re- cov ered, pr obabilistically , by reading at most t wo other co de p ositio ns. Such co des hav e numerous applications a nd co nstructions (with exp onential enco ding length) ar e known over ﬁnite ﬁelds of small characteristic. W e show that inﬁnite families of s uch linear 2-query LCCs do n ot exist over ﬁelds of characteristic z e ro o r la rge c hara cteristic regar dless o f the enco ding length. Generalization of kno wn results in com binatorial geome try . W e pro ve a qua ntit a- tive analo g of the Sylvester-Galla i theorem: Let v 1 , . . . , v m be a set of p oints in C d such that for every i ∈ [ m ] there exists at least δ m v a lues of j ∈ [ m ] suc h that the line thr o ugh v i , v j contains a third point in the s e t. W e show that the dimension of { v 1 , . . . , v m } is at most O (1 /δ 2 ). Our results genera liz e to the high dimensional case (replacing lines w ith planes, etc.) a nd to the case where the p oints a re color ed (as in the Motzkin-Rabin Theo rem). ∗ Microsof t Researc h N ew England. Email: boaz@microsoft.co m . Mo st of the work done while at Princeton Universit y an d supp orted by NS F grants CNS-0627526, CCF-042658 2 and CCF-0832797, and the P ack ard and Sloan fel low ships. † Department of Computer Science, Princeton Universit y . Email: zeev.dvir@gmai l.com . Researc h partially supp orted b y NSF grant CCF-0832797 and by t he Pac k ard fello wship. ‡ School of Mathematics, In stitute for Adv anced Study . Email: avi@ias.edu . Resear ch partially supp orted by NSF grants CCF-0832797 and DMS-0835373. § Department of Mathematics, T echnion - I IT. Email: amir.yehudayoff @gmail.com . Most of the work d one while at the IAS. R esearc h p artially supp orted by NS F grants CCF-0832797 and DMS-083537 3. 1 1 In tro duction In this work we study wh at c ombinatoria l prop erties of matrices guarante e high algebraic r ank , where a prop ert y is com bin atorial if it dep end s only on the zero/non-zero p attern of the matrix, and n ot on the v alues of its ent ries. This qu estion h as a rich history in mathematics (see Section 1.2), and some compu ter science motiv ations: Lo cally correctable codes. A lo c al ly c orr e ctable c o de is an error correcting co de in whic h for eve ry co deword y , giv en a corrupted version ˜ y of y and an index i , on e can reco ver the correct v alue of y i from ˜ y by lo oking only at v ery few co ord in ates of ˜ y . It is an op en question in co d ing theory to under s tand the tradeoﬀs b et ween the fraction of errors, lo calit y (n umber of co ordin ates read) and rate (ratio of message length to co deword length) of suc h co des, w ith v ery large gaps b et w een the k n o wn upp er b ounds and lo wer b ounds (see the surve y [T re04]). T he question is op en even for linear co d es, where the condition of b eing lo cally correctable turn s out to b e equiv alen t to the existence of low w eigh t co dewords in the dual co dewo rds that are “w ell-spread” in some precise tec hnical sense (see S ection 7). Because of the relation b et we en the rate of the co de and its du al, the question b ecomes equiv alen t to asking wh ether this com binatorial “w ell-spreadness” condition guaran tees high rank. Matrix rigidit y . A longstanding question is to come up w ith an explicit matrix that is rigid in the sens e th at its rank cannot b e reduced b y c hanging a small num b er of its en tries. Random matrices are extremely rigid, and suﬃ ciently go o d explicit constru ctions w ill yield lo wer b ounds for arithmetic circuits [V al7 7 ], though we are still ve ry far f rom ac h ieving this (see the surv ey [Lok09]). O ne ca n hop e that a com binatorial pr op erty guarantee ing large rank w ill b e r obust under small p erturbations, and hence a matrix satisfying su c h a prop erty w ill automaticall y b e rigid. In b oth these cases it is crucial to obtain bou n ds on th e rank that dep end solely on the zero/non-zero pattern of the matrix, without p lacing any restrictions on the n on -zero co eﬃcients. F or example, th er e are very str ong b ounds k n o wn for matrix rigidit y under the r estriction that the n on-zero co eﬃcien ts ha ve b ounded magnitude (see Chapter 3 in [Lok09]), b u t they only imply lo w er b ound s in a ve ry restricted m o del. In fact, there is a relat ion b et we en the t wo questions, and s uﬃcien tly go o d answe rs for the ﬁ rst question w ill imply answers for the second one [Dvi10]. W e stress that th ese t wo examples are in no wa y exhaustiv e. The interpla y b etw een com b inatorial and algebraic prop erties of matrices is a fascinating qu estion with man y p oten tial applications th at is still v ery p o orly un dersto o d. 1.1 Our Results In this work we giv e a com binatorial prop erty of complex m atrices that implies high rank. While not strong enough to p ro v e rigidit y r esults, we are able to us e it to obtain sev eral app lications in com binatorial geometry and lo cally correctable codes. Our m ain r esult is the follo wing the- orem, giving a lo wer b oun d on the rank of m atrix whose non-zero pattern f orms has certain 2 com b inatorial-design lik e prop erties in the sense that th e sets of non-zero ent ries in eac h column ha v e small in tersections. (This th eorem is restated as Theorem 3.2.) Theorem 1 (Rank b ound for design matrices) . L et m ≥ n . We say that an m × n c omplex matrix A is a ( q , k , t ) -design matrix if every r ow of A has at most q non-zer o entries, every c olumn of A has at le ast k non-zer o es entries, and the supp orts of every two c olumns interse ct in at most t r ows. F or every such A , r ank ( A ) ≥ n −  q · t · n 2 k  2 . W e also sho w that Theorem 1, and in fact any resu lt connecting the ze ro/non-zero p attern to r ank, can b e m ad e to h old ov er arbitrary c haracteristic zero ﬁelds and also o v er ﬁelds of suﬃcien tly large (dep ending on m, n ) ﬁnite c h aracteristic. 1.1.1 Applications to Com binat orial Geometry Our most immediate applications of Theorem 1 are to questions regarding line-p oin t incidences. Results on line-p oin t incidences hav e recen tly found use in the area of computational complexit y in relation to pseud o-randomness [BKT04, BIW06] and d e-randomization [KS09, SS10]. In this setting w e h a v e an arrangement of a ﬁnite num b er of p oin ts in real or complex space. Every suc h arrangemen t gives rise to a set of lines, namely , those lines that p ass thr ough at least t w o of the p oints in th e arrangemen t. Inf ormation ab out these lines can b e con v erted, in s ome cases, into information ab out the dimension of the set of p oin ts (i.e. the d im en sion of the space the p oin ts span). Our rank theorem can b e used to deriv e generalizati ons f or t wo w ell-kno wn theorems in this area: the Sylve ster-Gallai theorem an d the Motzkin-Rabin th eorem. Generalizing the Sylv ester-Gallai Theorem. The Sylvest er-Gallai (SG for short) th eorem sa ys that if m distinct p oin ts v 1 , . . . , v m ∈ R d are n ot collinear, then there exists a line that passes through exactly t wo of th em. In its con trap ositiv e form the SG theorem s a y s that if for ev ery i 6 = j the line through v i and v j passes thr ou gh a third p oint v k , then dim { v 1 , . . . , v m } ≤ 1, where dim { v 1 , . . . , v m } is the d imension of the s m allest aﬃne sub space con taining the p oin ts. This th eorem was ﬁr s t conjectured by Sylveste r in 1893 [Syl93], pro v ed (in du al form) by Melc hior in 1940 [Mel40], and then indep endent ly conjectured by Erdos in 1943 [Erd43] and p ro v ed b y Gallai in 1944. The SG theorem h as several b eautiful pro ofs and many generalizatio ns, s ee th e surve y [BM90 ]. Over the complex num b ers the (tight ) b ound on th e dimension is 2 instead of 1. The complex version w as ﬁrst pr o v en by Kelly [Kel86] using a deep r esults from algebraic geometry , and more recen tly , an eleme nta ry pro of was foun d by Elkies, Pretorius and Sw anep o el [ES06] who also pro v ed it o ve r the qu aternions with an upp er of 4 on the dimension. W e sa y that the p oin ts v 1 , . . . , v m (in R d or C d ) form a δ -SG conﬁguration if for ev ery i ∈ [ m ] there exists at least δ m v alues of j ∈ [ m ] such that the line thr ough v i , v j con tains a third p oint in the set. Szemeredi an d T rotter [S T 83] show ed that, when δ is large r than some absolute constan t close to 1, th en th e dimension of a δ -SG conﬁgur ation is at most one (o ver the reals). W e sho w the follo wing generalization of their r esult to arbitrary δ > 0 (and o v er the complex n umber s ). 3 Theorem 2 (Quan titativ e SG th eorem) . If v 1 , . . . , v m ∈ C d is a δ -SG c onﬁgur ation then dim { v 1 , . . . , v m } < 13 /δ 2 . W e note that one cannot replace the b ound 13 / δ 2 of Theorem 2 with 1 or ev en with an y ﬁxed constan t, as one can easily create a δ -SG conﬁ gu r ation of dimension r oughly 2 /δ b y placing the p oin ts on 1 /δ lines. This is analogo us to error correcting codes, where once the f r action δ of agreement b et ween the original and corrupted co dewo rd d rops b elo w half there can b e no unique d eco ding. In that sense our result can b e thought of as a list de c o ding v ariant of the S G theorem, whereas the result of [ST83] is its un ique deco d ing v arian t. W e also show an “a v erage case” v ersion of the SG theorem, pro ving a b oun d on the dimension of a large sub set of the p oints un der the assu mption that there are many collinear triples (see Theorem 4.8). W e also prov e a v ersion of Theorem 4.3 w ith lines replaced by k -ﬂats ( k -dimensional aﬃne subspaces). This generali zes a theorem of Hansen [Han65, BE67] which deals with the case α = 1. The state ment of this result is tec hnical and so w e giv e it in Section 5 w h ere it is also pro v en. Since our pro ofs u se elemen tary (and pu rely alge braic) redu ctions to the rank theorem, they hold o v er arbitrary ﬁelds of charact eristic zero or of suﬃcien tly large ﬁn ite c haracteristic. T his is in con trast to many of the kno w n pr o ofs of su c h theorems w h ic h often rely on sp eciﬁc prop erties of the real (or complex) num b ers. Ho we ve r, w e currently d o not reco ve r the full v ersion of the original SG th eorem, in th e sense that even for δ = 1 we d o not get a b ound of 1 (or 2 for complex n umbers) on the dimension. (Ho w ev er, the term 13 /δ 2 can b e impro ve d a bit in the δ = 1 case to obtain a b ound of 9 on the d imension.) Generalizing the Motzkin-Rabin Theorem. The Motzkin-Rabin (MR for sh ort) theorem (see e.g. [BM90]) is an in teresting v ariant of the Sylv ester-Gallai theorem that states that if p oints v 1 , . . . , v m ∈ R d are colored either r ed or blue and there is n o monochromatic line p assing through at least t wo p oin ts, then they are all co llinear. A s in the SG theorem, we obtain a quan titativ e generalization of the MR theorem suc h that (lett ing b and r b e the n um b ers of blue and red p oint s resp ectiv ely), if for ev ery b lue (resp. red) p oin t v , there are δ b blue (resp. δ r red) p oin ts v ′ where the line through v and v ′ passes through a red (resp. blu e) p oin t, then dim { v 1 , . . . , v m } ≤ O (1 /δ 4 ). W e also pro v e a th ree colors v arian t of the MR theorem, sho wing that if v 1 , . . . , v m are colored r ed, blue and green, an d all lines are not mono c hromatic, then dim { v 1 , . . . , v m } is at most some absolute constan t. 1.1.2 Lo cally Correctable Co des A (linear) q query lo cally correctable co de (( q , δ )-LCC for short) ov er a ﬁ eld F is a sub space C ⊆ F n suc h that, giv en an elemen t ˜ y th at disagrees with some y ∈ C in at m ost δ n p ositions and an index i ∈ [ n ], on e can reco v er y i with, sa y , probabilit y 0 . 9, by reading at most q coordinates of ˜ y . Ov er the ﬁ eld of t wo elements F 2 the standard Hadamard co d e construction yields a (2, δ )- query LCC with dimension Ω(log( n )) for constan t δ > 0 (see the surv ey [T re04]). In con trast w e show that for every constan t δ > 0 th ere do not exist inﬁnite family of suc h co d es o ver the complex num b ers: 4 Theorem 3 (Imp ossibilit y of 2-query LCCs o v er C ) . If C i s a 2 -query LCC for δ fr action of err ors over C , then dim( C ) ≤ O (1 /δ 9 ) . W e n ote that the Hadamard co nstru ction do es yield a lo c al ly de c o dable c o de o ver the co m- plex num b ers w ith dimension Ω(log n ). Lo cally deco dable co des are the r elaxatio n of a lo cally correctable codes where one only needs to b e able to reco ver the co ordin ates of the original message as opp osed to the co deword. Th us o ver the complex n u m b ers, th ere is a very strong separation b et ween the notions of lo cally d ecod able and lo cally correctable co des, w hereas it is consisten t with our knowledge that for, say , F 2 the rate/localit y tradeoﬀs of b oth n otions are the same. 1.2 Related W ork The idea to use matrix scaling to study str uctural pr op erties of matrices was already pr esen t in [CPR00]. Th is work, whic h w as also m otiv ated b y the problem of matrix rigidit y , studies the presence of short cycles in the graphs of non-zero entries of a square matrix. A related line of w ork on the rank of ‘design’ matrices is the w ork emerging f rom Hamada’s conjecture [Ham73]. (See [JT09] for a recen t result and more references.) Here, a design matrix is deﬁn ed using stricter conditions (eac h row/c olumn has exactly the same num b er of non-zeros and the inte rsections are also al l of the same s ize) whic h are more common in the literature dealing with com bin atorial designs. In order to b e completely consisten t with this line of work w e should hav e called our matrices ‘appro ximate-design’ matrices. W e chose to use the (already o verused) w ord ‘design’ to make the presenta tion more r eadable. W e also n ote that consider in g appro ximate d esigns only m ak es our resu lts stronger. Hamada’s conjecture states that of all zero/one matrices whose sup p ort comes from a design (in the stricter sense), th e minimal rank is obtained b y matrices coming from geometric designs (in our language, Reed-Muller co des). In contrast to this pap er, the emph asis in this line of works is typica lly on small ﬁnite ﬁelds. W e note here that th e connection b et ween Hamada’s conjecture and LC Cs wa s already observ ed by Bark ol, Ishai and W einreb [BIW07] who also conjectured (ov er small ﬁelds ) the ‘approxima te- design’ versions w hic h we pro ve her e for large ﬁ elds. Another p lace where the supp ort of a matrix is co nnected to its r an k is in grap h theory where we are inte rested in m inimizing the rank of a (square, s ymmetric) real matrix whic h has the same supp ort as the adjacency matrix of a giv en graph. This line of w ork go es bac k for ov er ﬁft y y ears and has m an y applications in graph theory . S ee [FH07] for a r ecen t su rve y on this topic. Ov er the reals we ca n also ask ab ou t the min im al rank of matrices with certain sign-p attern . That is, giv en a matrix o v er { 1 , − 1 } , wh at is the minimal rank of a matrix w hic h h as the same sign-pattern. This minimal rank is calle d the sign-r ank of a matrix. The question of coming up with (combinato rial or otherwise) prop erties that imply high sign-rank is one of ma jor imp ortance a nd has strong connectio ns to comm unication complexit y , learning theory and circuit complexit y , among others. F or a recen t w ork with plen t y of references see [RS08]. In particular w e wo uld lik e to mentio n a conn ection to the wo rk of F orster [F or02] on the sign-rank of the Hadamard matrix. (An earlier ve rsion of this w ork u sed a v ariant [Bar98, Har10] of a lemma f r om [F or02] instead of the results of [RS89] on matrix scaling to obtain our main result.) 5 1.3 Organization In Section 2 w e give a high level o ve rview of our tec hniques. In Section 3 w e p ro v e our main result on the rank of d esign matrices. In Section 4 w e pro v e our quantit ativ e v arian ts of the Sylv ester-Gallai theorem. In Section 5 we prov e the h igh-dimensional analog of Theorem 4.3 where lines are replaced with ﬂats. In Section 6 w e pro ve our generalizatio ns of the Motzkin- Rabin theorem. In Section 7 we pr o v e our results on lo cally correctable co des. In Section 8 w e sho w h o w our results extend to other ﬁelds. W e conclude in S ection 9 with a discussion of op en problems. 2 Our T ec hniques W e now giv e high-lev el pro of o ve rviews for some of our r esu lts. 2.1 Rank Low er Bounds for Design Matrices Theorem 1 – the rank lo wer b ound for design matrices – is pr ov ed in tw o steps. W e no w sk etc h the p ro of, ignoring some subtleties and optimizations. The pr o of starts with the observ ation that, as in the case of matrix rigidit y and similar questions, the result is muc h easier to pr o v e giv en a b ound on the magnitude of the non-zero en tries. Indeed, if A is a ( q , k , t )-design m atrix and all of its non-zero en tries ha v e absolute v alue in [1 /c, 1] for some constant c , then the n × n matrix M = A ∗ A is diagonal ly dominant , in the s en se that for all i 6 = j , m ii ≥ k /c 2 but | m ij | ≤ t . (Here A ∗ denotes the conjugate transp ose of A .) Thus one can use kno wn results on suc h matrices (e.g. [Alo09]) to argue th at r ank( A ) ≥ rank( M ) ≥ n − ( nt c 2 /k ) 2 . Our main idea is to reduce to this case where the n on -zero co eﬃcien ts of A are (roughly) b ounded using matrix sc aling . A sc aling ˆ A of a matrix A is obtained b y multiplying f or all i, j , the i ’th row of A b y some p ositiv e num b er ρ i and the j ’th column of A by some p ositiv e num b er γ j . Clearly , A and ˆ A share the same rank and zero/non-zero pattern. W e u se kn o wn matrix-scali ng results [Sin64, RS89] to sho w that eve ry ( q , k , t )-design matrix A has a scaling in whic h eve ry en try has magnitud e at most (roughly) 1 but its columns ha v e n orm at least (r ou gh ly) p k /q . W e n ote that the typical application of matrix-scaling w as with resp ect to th e ℓ 1 -norm of the ro ws and columns. He re we tak e a diﬀeren t p ath: W e use scaling w ith resp ect to ℓ 2 -norm. W e d efer th e d escription of this step to Section 3 but the high leve l idea is to use a theorem of [RS89] that sho ws th at s uc h a scaling exists (in fact without the dep endence on q ) if A had the prop erty of not con taining any large all-zero su b-matrix. While this prop ert y cannot b e in general guarant eed, w e sho w that b y r ep eating some r o ws of A one can obtain a matrix B that has this prop erty , and a scaling of B can b e con ve rted in to a scaling of A . Since our lo w er b ound on the en try m ii in the b ounded co eﬃcient case (wh ere aga in M = A ∗ A ) only u sed the fact that the columns hav e large norms, w e can u se the same argument as ab ov e to lo w er b oun d the rank of M , and hence of A . 6 2.2 Generalized Sylv ester-Gallai Theorem Recall that th e qu an titativ e SG theorem (Theorem 2) states that every δ -SG conﬁguration v 1 , . . . , v n , has dimension at most 13 /δ 2 . Our pro of of Theorem 2 uses Theorem 1 as follo ws. Supp ose for starters that every one of these lines p assed through exactly three p oints. Eac h suc h line induces an equation of the form αv i + β v j + γ v k = 0. No w for m = δ n 2 , let A b e the m × n matrix whose ro w s corr esp ond to these equatio ns. S ince ev er y tw o p oin ts participate in only one line, A will b e a (3 , δ n, 1) design matrix, meaning that according to Theorem 1, A ’s rank is at least n −  3 2 δ  2 . Since A times the matrix whose ro w s are v 1 , . . . , v n is zero we ha v e dim { v 1 , . . . , v n } ≤ n − rank( A ). W e th u s get an up p er b oun d of ⌊ 9 / 4 ⌋ = 2 on this dimension. T o h andle the case when some lines con tain more than three p oints, we c ho ose in some careful w a y f rom eac h line ℓ con taining r p oin ts a sub s et of the  r 3  equations of the form ab o ve th at it induces on its p oints. W e sho w that at some sm all loss in th e parameters w e can still ensu re the set of equations f orm s a design, hence again deriving a lo w er b ound on its r ank via Theorem 1. Our metho d extend also to an “a verag e case” SG th eorem (Theorem 4.8), where one only requires that the set of p oin ts supp orts man y (i.e., Ω( n 2 )) co llinear triples and that eac h pair of p oint s app ear together in a few collinear triples. In this case we are able to s h o w that there is a subset of Ω( n ) p oin ts whose span has d imension O (1). S ee Section 4 for more details. Our generalizat ions of the Motzkin-Rabin theorem follo w from our theorem on δ -SG conﬁ gurations via simple r eductions (see Section 6). 2.3 Lo cally Correct able Co des A t ﬁrst sight , Theorem 3 – non existence of 2 query locally correctable co des o ver C – seems lik e it should b e an immediate corolla ry of Theorem 2 . Supp ose that a code C m ap s C d to C n , and let v 1 , . . . , v n denote the ro ws of its generating matrix. That is, the co de maps a message x ∈ C d to the v ector ( h v 1 , x i , . . . , h v n , x i ). The f act that C is a 2 query LCC for δ errors implies that for every such row v i , there are roughly δ n pairs j, k such that v i is in the span of { v j , v k } . Using some simple scali ng/c hange of basis, this giv es precisely the co ndition of b eing a δ -SG conﬁgur ation, sa v e for one ca ve at: In a code there is no guarantee that all the ve ctors v 1 , . . . , v n are distinct. That is, the co d e ma y ha v e rep eated co ord in ates that are alw a ys identical . In tuitiv ely it seems that such rep etitions should not help at all in constr u cting LCCs but pr o vin g this tur n ed out to b e elusiv e. In fact, our p ro of of Theorem 3 is rather more complicated than the pro of Theorem 2, in vo lving rep eated applications of Theorem 1 wh ic h result also in somewhat p o orer quan titativ e b ound s. Th e idea b ehind the pro of to use a v ariant of the “a ve rage case” SG theorem to r ep eatedly ﬁnd Ω( n ) p oint s among v 1 , . . . , v n whose span has O (1) dimension, u n til there are no more p oin ts left. W e defer all details to Section 7. Giv en Th eorem 1, on e ma y ha v e exp ected that Th eorem 3 co uld b e extended for LCC s of an y constan t n um b er q of queries. After all, the cond ition of C b eing an LCC intuitiv ely seems lik e only a s light relaxation of requirin g that the dual co de of C has a generating matrix whose non-zero pattern is a com binatorial design, and indeed in kno wn constru ctions of LCCs, the dual code do es form a design. W e are not, ho wev er, able to extend our results to 3 and more queries. A partial explanat ion to our inabilit y is that 3 query LCCs giv e rise to co nﬁgur ation of planes (instead of lines) and p oint and planes exhibit m uch more complicated com binatorial 7 prop erties th an lines. 3 Rank of Design Matrices In this secti on w e p ro v e our main resu lt wh ic h giv es a lo w er b ound on th e rank of matrice s whose zero/non-zero patte rn satisﬁes certain p rop erties. W e start b y deﬁn ing these p r op erties formally . Deﬁnition 3.1 (D esign m atrix) . Let A b e an m × n matrix o ver some ﬁeld. F or i ∈ [ m ] let R i ⊂ [ n ] denote the set of indices of all non-zero entries in the i ’th r o w of A . Similarly , let C j ⊂ [ m ], j ∈ [ n ], denote the set of non-zero indices in the j ’th column. W e say that A is a ( q , k , t ) -design matrix if 1. F or all i ∈ [ m ], | R i | ≤ q . 2. F or all j ∈ [ n ], | C j | ≥ k . 3. F or all j 1 6 = j 2 ∈ [ n ], | C j 1 ∩ C j 2 | ≤ t . Theorem 3.2 (Restateme nt of Theorem 1 – rank of d esign matrices) . L e t A b e an m × n c omplex matrix. If A is a ( q, k , t ) -design matrix then r ank ( A ) ≥ n −  q · t · n 2 k  2 . Remark 3.3. T he pro of of the theorem actually holds und er a s lightly weak er co ndition on the sizes of th e intersecti ons. In stead of requiring that | C j 1 ∩ C j 2 | ≤ t for all p airs of columns j 1 6 = j 2 , it is enough to ask that X j 1 6 = j 2 | C j 1 ∩ C j 2 | 2 ≤ n 2 · t 2 . That is, there could b e some pairs with large intersect ion as long as the a v erage of the squ ares is not too large. The pr o of of the theorem is giv en b elo w, follo wing some preliminaries. 3.1 Preliminaries for the Pro of of Theorem 3.2 Notation: F or a set of r eal v ectors V ∈ C n w e denote by rank( V ) the dimension of the v ector space sp an n ed by elemen ts of V . W e den ote the ℓ 2 -norm of a v ector v b y k v k . W e d enote b y I n the n × n iden tit y matrix. W e start with deﬁnitions an d results on matrix scaling. Deﬁnition 3.4. [Matrix scaling] Let A b e an m × n complex matrix. Let ρ ∈ C m , γ ∈ C n b e t w o complex ve ctors with all entries non-zero. W e denote by SC ( A, ρ, γ ) 8 the matrix obtained from A b y multiplying the ( i, j )’th elemen t of A by ρ i · γ j . W e say that t w o matrices A, B of the same dimensions are a s caling of eac h other if th er e exist n on -zero vect ors ρ, γ s uc h that B = SC ( A, ρ, γ ). It is easy to c hec k that this is an equ iv alence r elation. W e refer to the ele ments of th e v ector ρ as the r ow sc aling c o eﬃcients and to the elemen ts of γ as the c olumn sc aling c o eﬃcients . Notice that t wo matrices wh ic h are a scaling of eac h other hav e the same rank and the same pattern of zero and n on -zero en tries. Matrix scaling originated in a pap er of Sinkh orn [Sin64] and h as b een widely stu died since (see [LS W00] for more bac kgroun d ). The follo wing is a sp ecial case of a theorem from [RS89] that giv es suﬃcien t cond itions f or ﬁ nding a scaling of a matrix which h as certain row and column sums. Deﬁnition 3.5 (Prop erty- S ) . Let A b e an m × n matrix o ver some ﬁeld. W e sa y that A satisﬁes Pr op e rty- S if for ev ery zero sub -matrix of A of size a × b it holds that a m + b n ≤ 1 . (1) Theorem 3.6 (Matrix scaling theorem, Theorem 3 in [RS89] ) . L et A b e an m × n r e al matrix with non-ne gative entries which satisﬁes Pr op e rty- S . Then, for every ǫ > 0 , ther e exists a sc aling A ′ of A such that the sum of e ach r ow of A ′ is at most 1 + ǫ and the sum of e ach c olumn of A ′ is at le ast m/n − ǫ . Mor e over, the sc aling c o eﬃcients use d to obtain A ′ ar e al l p ositive r e al numb ers. The pro of of the th eorem is algorithmic [Sin64]: Start by normalizing A ’s ro ws to h a v e su m 1, then normalize A ’s columns to hav e s um m/n , then go bac k to normalizing the rows th e ha v e sum 1, and s o forth. I t ca n be sh o w n (using a suitable p oten tial function) that this pro cess ev en tually transforms A to the claimed form (since A has Prop ert y- S ). W e will us e the follo wing easy corollary of the ab o ve theorem. Corollary 3.7 ( ℓ 2 2 -scaling) . L et A = ( a ij ) b e an m × n c omplex matr ix which satisﬁes Pr op erty- S . Then, for every ǫ > 0 , ther e exists a sc aling A ′ of A such that for every i ∈ [ m ] X j ∈ [ n ] | a ij | 2 ≤ 1 + ǫ and for every j ∈ [ n ] X i ∈ [ m ] | a ij | 2 ≥ m/n − ǫ. Pr o of. Let B = ( b ij ) = ( | a ij | 2 ). Then B is a r eal non-negativ e matrix satisfying Pr op ert y- S . Applying Theorem 3.6 we get that for all ǫ > 0 there exists a scaling B ′ = SC ( B , ρ, γ ), with ρ, γ p ositiv e real v ectors, whic h h as ro w sums at most 1 + ǫ and column su ms at least m /n − ǫ . Letting ρ ′ i = √ ρ i and γ ′ i = √ γ i w e get a scaling SC ( A, ρ ′ , γ ′ ) of A with the r equired prop erties. W e will u se a v ariant of a w ell kno wn lemma (see for example [Alo09]) w hic h p r o vides a b ound on the rank of matrices whose d iagonal en tr ies are muc h larger than the oﬀ-diagonal ones. 9 Lemma 3.8. L et A = ( a ij ) b e an n × n c omplex hermitian matrix and let 0 < ℓ < L b e inte gers. Supp ose that a ii ≥ L for al l i ∈ [ n ] and tha t | a ij | ≤ ℓ for al l i 6 = j . Then r ank ( A ) ≥ n 1 + n · ( ℓ/L ) 2 ≥ n − ( nℓ/L ) 2 . Pr o of. W e can assum e w.l.o.g. that a ii = L for all i . If not, then w e can make the inequalit y in to an equalit y by m ultiplying the i ’th row and column by ( L/a ii ) 1 / 2 < 1 without changing the r ank or br eaking the symmetry . Let r = rank( A ) and let λ 1 , . . . , λ r denote the non -zero eigen v alues of A (coun ting multiplic ities). Since A is hermitian we ha v e that the λ i ’s are real. W e hav e n 2 · L 2 = tr( A ) 2 = r X i =1 λ i ! 2 ≤ r · r X i =1 λ 2 i = r · n X i,j =1 | a ij | 2 ≤ r · ( n · L 2 + n 2 · ℓ 2 ) . Rearranging w e get the required b ound. The second inequ alit y in the statement of the lemma follo ws from th e fact that 1 / (1 + x ) ≥ 1 − x for all x . 3.2 Pro of of Theorem 3.2 T o p ro v e the theorem w e will ﬁrs t ﬁnd a scaling of A so that the norm s (squared) of the columns are large and suc h th at eac h en try is small. Our ﬁrst step is to ﬁnd an n k × n matrix B that w ill satisfy Prop ert y- S and will b e comp osed from rows of A s.t. eac h ro w is rep eated with multiplicit y b et ween 0 and q . T o ac hiev e this we will describ e an alg orithm that bu ilds the matrix B iterativ ely by concatenating to it ro ws from A . The algorithm will mark entries of A as it co ntin ues to add ro w s . Keeping trac k of these marks will help u s decide whic h r o ws to add next. Initially all the en tries of A are unmarke d . The alg orithm pr o ceeds in k steps. At step i ( i goes from 1 to k ) the algorithm p ic ks n ro ws from A and adds them to B . These n ro ws are c hosen as follo ws: F or ev ery j ∈ { 1 , . . . , n } pick a ro w that h as an un mark ed non-zero entry in the j ’th column and mark this non-zero en try . The reason why such a row exists at al l steps is that ea c h column con tains at least k non-zero en tries, and in eac h step we m ark at most one n on-zero ent ry in eac h column . Claim 3.9. The matrix B obta ine d by the algorithm has Pr op erty- S and e ach r ow of A is adde d to B at most q times. Pr o of. The n rows added at eac h of the k steps f orm an n × n matrix w ith non-zero diagonal. Th us they satisfy Prop ert y- S . It is an easy exercise to verify that a concatenation of matrices with Prop erty- S also h as this prop ert y . The b ound on th e num b er of times eac h ro w is added to B follo ws from the fact that eac h ro w h as at most q non-zero en tries and eac h time w e add a row to B w e mark one of its non-zero en tries. Our next step is to obtain a scaling of B and , from it, a s caling of A . Fix some ǫ > 0 (which will later tend to zero). Applying Corollary 3.7 we get a scaling B ′ of B such that the ℓ 2 -norm 10 of eac h row is at most √ 1 + ǫ and the ℓ 2 -norm of eac h column is at least p nk /n − ǫ = √ k − ǫ . W e now obtain a scaling A ′ of A as follo ws: The scaling of the columns are th e same as for B ′ . F or the ro ws of A app earing in B we tak e the maximal scaling co eﬃcien t used for these r o w s in B ′ , that is, if ro w i in A app ears as ro ws i 1 , i 2 , . . . , i q ′ in B , then the scaling co eﬃcien t of ro w i in A ′ is the maximal scaling co eﬃcien t of rows i 1 , i 2 , . . . , i q ′ in B ′ . F or ro ws not in B , w e p ic k scaling co eﬃcien ts so that their ℓ 2 norm (in the ﬁnal s caling) is equal to 1. Claim 3.10. The matrix A ′ is a sc aling of A such that e ach r ow has ℓ 2 -norm at most √ 1 + ǫ and e ach c olumn has ℓ 2 -norm at le ast p ( k − ǫ ) /q . Pr o of. The fact that th e r ow norms are at most √ 1 + ǫ is trivial. T o argue ab ou t the column norms obser ve that a column of B ′ is obtained from rep eating eac h non-zero elemen t in the corresp ondin g column of A ′ at most q times (together with some zeros). Therefore, if w e denote b y c 1 , . . . , c s the n on -zero en tries in some column of A ′ , we ha v e th at s X i =1 m i · | c i | 2 ≥ k − ǫ, where the m i ’s are int egers b etw een 0 and q . In th is last inequality we also relied on the fact that we chose the maximal ro w scaling co eﬃcien t among all those that corresp ond to the same ro w in A . T h erefore, s X i =1 | c i | 2 ≥ ( k − ǫ ) /q , as requ ir ed. Our ﬁn al step is to argue ab out the rank of A ′ (whic h is the same as the r ank of A ). T o this end, consider the matrix M = ( A ′ ) ∗ · A ′ , where ( A ′ ) ∗ is A ′ transp osed conjugate. Then M = ( m ij ) is an n × n h ermitian matrix. Th e diagonal entries of M are exactly th e squares of the ℓ 2 -norm of the columns of A ′ . Th erefore, m ii ≥ ( k − ǫ ) /q for all i ∈ [ n ]. W e now upp er b ound the oﬀ-diagonal ent ries. The oﬀ-dia gonal en tries of M are the in n er pro du cts of d iﬀeren t columns of A ′ . T he intersect ion of the supp ort of eac h pair of d iﬀeren t columns is at most t . The norm of eac h ro w is at most √ 1 + ǫ . F or ev ery t w o real num b ers α, β so that α 2 + β 2 ≤ 1 + ǫ we h a v e | α · β | ≤ 1 / 2 + ǫ ′ , where ǫ ′ tends to zero as ǫ tends to zero. Therefore | m ij | ≤ t · (1 / 2 + ǫ ′ ) for all i 6 = j ∈ [ n ]. Applying Lemm a 3.8 we get that rank( A ) = rank( A ′ ) ≥ n −  q · t (1 / 2 + ǫ ′ ) · n k − ǫ  2 . Since th is h olds for all ǫ > 0 it holds also for ǫ = 0, wh ich giv es th e requ ired b oun d on the r ank of A . 11 4 Sylv ester-Gallai Conﬁgurations In this section we p ro v e the quantit ativ e S ylv ester-Gallai (SG) Theorem. W e will b e in terested with point conﬁgurations in real and complex space. These are ﬁnite sets of distinct p oin ts v 1 , . . . , v n in R d or C d . Th e dimension of a conﬁguration is deﬁned to b e the dimension of the smallest aﬃne subspace con taining all p oin ts. Deﬁnition 4.1 (Sp ecial and ordinary lines) . Let v 1 , . . . , v n ∈ C d b e a set of n distinct p oints in d -dimensional complex space. A line ℓ passing throu gh at least three of these p oin ts is called a sp e cial line. A line p assin g through exactly t w o p oints is called an or dinary line. Deﬁnition 4.2 ( δ -SG conﬁguration) . Let δ ∈ [0 , 1]. A set of n distinct p oin ts v 1 , . . . , v n ∈ C d is called a δ -SG c onﬁgur ation if for ev ery i ∈ [ n ], there exists a family of sp ecial lines L i all passing through v i and at least δ n of th e p oin ts v 1 , . . . , v n are on the lines in L i . (Note that eac h collection L i ma y co ve r a diﬀeren t su bset of the n p oints.) The main result of th is section bou n ds the dimension of δ -SG conﬁgurations for all δ > 0. Since w e can alwa ys satisfy the deﬁnition by spreading the p oints evenly o v er 1 /δ lines we k n o w that the dimension can b e at least 2 /δ (and in fact in complex space at least 3 /δ ). W e p ro v e an upp er b oun d of O (1 /δ 2 ). Theorem 4.3 (Restatemen t of Theorem 2 – quan titativ e S G theorem) . L et δ ∈ (0 , 1] . L et v 1 , . . . , v n ∈ C d b e a δ -SG c onﬁgur ation. Then dim { v 1 , . . . , v n } < 13 / δ 2 . Mor e over, the dimension of a 1 -SG c onﬁgur ation is at most 10 . The constant s in the pro of hav e b een optimized to the b est of our abilities. Notice that in the ab o v e theorem δ can b e dep end an t on n . F or example, a (1 / log ( n ))-SG conﬁguration of n p oints can ha v e rank at most O (log ( n ) 2 ). 4.1 Preliminaries to t he Pro of of Theorem 4.3 The notion of a latin square will turn out u seful in the pro of: Deﬁnition 4.4 (Lat in sq u ares) . An r × r latin squar e is an r × r m atrix D suc h that D i,j ∈ [ r ] for all i, j and ev ery num b er in [ r ] app ears exactly once in eac h ro w and in eac h column. A latin square D is called diagonal if D i,i = i for all i ∈ [ r ]. Theorem 4.5 ([Hil73 ]) . F or every r ≥ 3 ther e exists a diagonal r × r latin squar e. W e note that w e use diagonal latin squares only to optimize constan t factors. If one do es not care ab out such factors then there is a simp le construction th at serves th e same goal. The follo wing lemma is an easy consequence of the ab o v e th eorem. Lemma 4.6. L et r ≥ 3 . Then ther e exists a set T ⊂ [ r ] 3 of r 2 − r triples that satisﬁes the fol lowing pr op erties: 12 1. Each triple ( t 1 , t 2 , t 3 ) ∈ T is of thr e e distinct elements. 2. F or e ach i ∈ [ r ] ther e ar e exactly 3( r − 1) triples in T c ontaining i as an e lement. 3. F or every p air i, j ∈ [ r ] of distinct elements ther e ar e at most 6 triples in T which c ontain b oth i and j as elements. Pr o of. Let D b e an r × r diagonal latin squ are which we kno w exists fr om Theorem 4.5 . De ﬁne T ⊂ [ r ] 3 to b e the set of all triples ( i, j, k ) ∈ [ r ] 3 with i 6 = j such that D i,j = k . The n umber of suc h trip les is r 2 − r . Prop erty 1 h olds by the deﬁn ition of diagonal latin square— w e ca nn ot ha v e D i,j = i for j 6 = i since D i,i = i and ev ery r o w in D has d istinct as th e ( i, i ) en try in D is lab eled i for all i ∈ [ r ], and similarly we cannot h av e D i,j = j for i 6 = j . Let i ∈ [ r ]. By construction, ther e are r − 1 triples in T whic h h av e i as their ﬁrst entry , and r − 1 triples that h a v e i as their second en try . There are also r − 1 triples in T w hic h h av e i as their last en try , s in ce for ev ery one of the r − 1 ro ws i ′ 6 = i there is exactly one lo cation j ′ 6 = i ′ in whic h the lab el i app ears, and th at con tributes the triple ( i ′ , j ′ , i ) to T . This pr o v es Prop ert y 2. T o p r o v e Prop ert y 3 observe that t wo triples in T ca n agree in at m ost one place. F or example, kno wing the ro w and co lumn determines the label, kno wing the row and lab el determines the column, and so forth. Th erefore, a pair ( i, j ) cannot app ear in m ore than 6 triples s ince otherwise there w ould ha ve b een at least tw o triples with i, j at the same places, and th ese triples w ould violate the ab o v e rule. 4.2 Pro of of Theorem 4.3 Let V b e the n × d matrix whose i ’th ro w is the v ector v i . Assume w.l.o.g. that v 1 = 0. Th us dim { v 1 , . . . , v n } = rank( V ) . The o v erview of the proof is as follo ws. W e w ill ﬁrst build an m × n matrix A that will satisfy A · V = 0. Th en, we will argu e that the rank of A is large b ecause it is a d esign matrix. This will s ho w that the rank of V is small. Consider a sp ecial line ℓ which p asses th rough thr ee p oin ts v i , v j , v k . Th is giv es a linear dep end ency among the three v ectors v i , v j , v k (w e id entify a p oin t with its v ector of co ordinates in the standard basis). In other w ords, th is giv es a vecto r a = ( a 1 , . . . , a n ) which is non-zero only in the th ree co ordinates i, j, k and such th at a · V = 0. I f a is n ot unique, c ho ose an arbitrary v ector a with these prop erties. Our strategy is to p ic k a f amily of collinear triples among the p oints in our conﬁguration and to build the matrix A from ro ws corresp ondin g to these trip les in the ab ov e manner. Let L denote the set of all sp ecial lines in the conﬁgur ation (i.e. all lines con taining at least three p oin ts). Th en eac h L i is a subset of L con taining lines passing through v i . F or eac h ℓ ∈ L let V ℓ denote the set of p oin ts in the conﬁguration w hic h lie on the line ℓ . Then | V ℓ | ≥ 3 and we can assign to it a family of triples T ℓ ⊂ V 3 ℓ , giv en by Lemma 4.6 (we id en tify V ℓ with [ r ], where r = | V ℓ | in some arbitrary w a y). 13 W e no w construct the matrix A b y going ov er all lines ℓ ∈ L and for eac h trip le in T ℓ adding as a ro w of A the v ector with three non-zero coeﬃcient s a = ( a 1 , . . . , a n ) describ ed ab o v e (so that a is the linear dep end en cy b et ween the three p oints in the triple). Since the matrix A sati sﬁes A · V = 0 b y construction, we only hav e to argue that A is a design matrix and b ound its rank. Claim 4.7. The matrix A is a (3 , 3 k , 6) -design matrix, wher e k , ⌊ δn ⌋ − 1 . Pr o of. By construction, eac h ro w of A has exactly 3 non-zero ent ries. The n umb er of non-zero en tries in column i of A corresp onds to the n umber of triples we used th at con tain the p oin t v i . These can come from all s p ecial lines conta ining v i . S u pp ose there are s sp ecial lines con taining v i and let r 1 , . . . , r s denote the n u m b er of p oin ts on eac h of those lines. Then, since the lines through v i ha v e only the p oin t v i in common, w e hav e th at s X j =1 ( r j − 1) ≥ k . The pr op erties of the families of trip les T ℓ guaran tee th at th ere are 3( r j − 1) triples cont aining v i coming from the j ’th line. Therefore there are at least 3 k triples in total conta ining v i . The size of the in tersection of column s i 1 and i 2 is equ al to the num b er of tr iples con taining the p oint s v i 1 , v i 2 that w ere used in the constru ction of A . These triples can only come fr om one sp ecial line (the line con taining th ese t wo p oin ts) and so, by Lemma 4.6 , there can b e at most 6 of those. Applying Th eorem 3.2 we get that rank( A ) ≥ n −  3 · 6 · n 2 · 3 k  2 ≥ n −  3 · n δ n − 2  2 ≥ n −  3 · n · 13 11 · δ n  2 > n − 13 /δ 2 , where the third inequality holds as δ n ≥ 13 s in ce otherwise the theorem trivially holds. Since A · V = 0 we h a v e that rank( A ) + rank( V ) ≤ n. This imp lies that rank( V ) < 13 /δ 2 , whic h completes the pro of. F or δ = 1, the calculation ab ov e y ields rank( V ) < 11. 4.3 Av erage-Case V ersion In this section w e use Th eorem 4.3 to argue ab out th e case where we only kno w that there are man y collinear triples in a conﬁguration. 14 Theorem 4.8 (A ve rage-case SG th eorem) . L et V = { v 1 , . . . , v m } ⊂ C d b e a set of m distinct p oints. L e t T b e the set of (unor der e d) c ol line ar triples in V . Supp ose | T | ≥ αm 2 and that every two p oints v , v ′ in V app e ar i n at most c triples in T , then ther e exists a subset V ′ ⊂ V such that | V ′ | ≥ αm/ (2 c ) and dim ( V ′ ) ≤ O (1 /α 2 ) . Notice that the b ound on the num b er of triples con taining a ﬁxed p air of p oin ts is necessary for the theorem to hold. If w e remo ve this assumption than w e could create a coun ter-example b y arranging the p oin ts so that m 2 / 3 of them are on a line and the rest sp an the entire space. Lemma 4.9. L et H b e a 3 -r e gu lar hyp er gr aph with vertex set [ m ] and αm 2 e dges of c o- de gr e e at most c (i.e. for every i 6 = j in [ m ] , the set { i, j } is c ontaine d in at most c e dges). Then ther e is a subset M ⊆ [ m ] of size | M | ≥ αm/ (2 c ) so that the minimal de gr e e of the sub-gr aph of H induc e d by M is at le ast αm/ 2 . Pr o of. W e describ e an iterativ e pro cess to ﬁnd M . W e start with M = [ m ]. While there exists a ve rtex of degree less than αm/ 2, remo v e this v er tex from M and remov e all edges con taining this ve rtex from H . Con tinuing in this fashion w e conclude with a set M suc h that every p oint in M has degree at least αm/ 2. This pr o cess remov ed in total at most m · αm/ 2 edges and th us the new H still con tains at least αm 2 / 2 ed ges. As th e co-degree is at most c , every v ertex app ears in at most cm edges. Th u s, the size of M is of size at least αm/ (2 c ). Pr o of of The or em 4.8. The family of triples T deﬁnes a 3-regular hyp ergraph on V of co-degree at most c . Lemma 4.9 thus imp lies that there is a subset V ′ ⊆ V of size | V ′ | ≥ αm/ (2 c ) that is an ( α/ 2)-SG conﬁguration. By Theorem 4.3, V ′ has dimension at most O (1 /α 2 ). 5 Robust SG Theorem for k -Flats In this section w e pro ve tw o high-dimensional analo gs of the SG theorem. Let ﬂ( v 1 , . . . , v k ) (ﬂ for ‘ﬂat’) denote th e aﬃne span of k p oint s (i.e. the p oints that can b e written as linear com b inations with co eﬃcient s that sum to one). W e call v 1 , . . . , v k indep endent if their ﬂat is of dimension k − 1 (dimension means aﬃne dimension), and sa y that v 1 , . . . , v k are dep endent otherwise. A k -ﬂat is an aﬃne sub space of d imension k . In the follo win g V is a set of n distinct p oin ts in complex sp ace C d . A k -ﬂat is called or dinary if its intersec tion with V is contai ned in the union of a ( k − 1)-ﬂat and a single p oint. A k -ﬂat is elementary if its int ersection with V h as exac tly k + 1 p oin ts. Notice that for k = 1 (lines) the tw o notions of ordinary and element ary coincide. F or dimen sions higher than one, there are tw o diﬀerent deﬁnitions that generalize that of SG conﬁguration. Th e ﬁrst deﬁn ition is b ased on ordin ary k -ﬂats (though in a sligh tly stronger w a y whic h w ill b e more useful in the pro ofs to come). The second deﬁnition (wh ic h is less restricted than the ﬁrst one) uses elemen tary k -ﬂ ats. Deﬁnition 5.1. The set V is a δ -SG ∗ k conﬁguration if for every indep enden t v 1 , . . . , v k ∈ V there are at least δn p oin ts u ∈ V s.t. either u ∈ ﬂ( v 1 , . . . , v k ) or the k -ﬂat ﬂ( v 1 , . . . , v k , u ) con tains a p oin t w outside ﬂ ( v 1 , . . . , v k ) ∪ { u } . 15 Deﬁnition 5.2. The set V is a δ -SG k conﬁguration if for every indep enden t v 1 , . . . , v k ∈ V there are at least δ n p oints u ∈ V s.t. either u ∈ ﬂ( v 1 , . . . , v k ) or the k -ﬂat ﬂ( v 1 , . . . , v k , u ) is not element ary . Both deﬁn itions coincide w ith th at of S G conﬁguration wh en k = 1: Indeed, ﬂ( v 1 ) = v 1 and ﬂ( v 1 , u ) is the line through v 1 , u . T herefore, u is never in ﬂ( v 1 ) and the line ﬂ( v 1 , u ) is not elemen tary iﬀ it con tains at least one p oint w 6∈ { v 1 , u } . W e prov e t wo high-dimensional v ersions of the SG theorem, eac h corresp onding to one of the deﬁnitions ab ov e. Th e ﬁ rst uses the more restricted ‘star’ d eﬁnition and gives a strong upp er b ound on dimension. The second uses the less restricted deﬁnition and giv es a weak er b ound on dimension. Theorem 5.3. L et V b e a δ -SG ∗ k c onﬁgur ation. Then dim ( V ) ≤ f ( δ , k ) with f ( δ , k ) = O  ( k /δ ) 2  . Theorem 5.4. L et V b e a δ -SG k c onﬁgur ation. Then dim ( V ) ≤ g ( δ , k ) with g ( δ, k ) = 2 C k /δ 2 with C > 1 a universal c onstant. The pr o ofs of the tw o theorems are b elo w . Th eorem 5.3 follo ws by an appropriate indu ction on the dimension, u s ing the (one-dimensional) robust SG theorem. Th eorem 5.4 follo w s by reduction to T heorem 5.3. Before pro vin g the theorems we set some notations. Fix some p oin t v 0 ∈ V . By a normaliza- tion w.r.t. v 0 w e m ean an aﬃne transform ation N : C d 7→ C d whic h ﬁrst mo v es v 0 to zero, th en pic ks a hyp erplane H s.t. no p oint in V (after the sh if t) is p arallel to H (i.e h as inner pro duct zero with the orthogonal v ector to H ) and ﬁnally m ultiplies eac h p oin t (other than zero) by a constan t s.t. it is in H . Claim 5.5. F or such a mapping N we have that v 0 , v 1 , . . . , v k ar e dep endent iﬀ N ( v 1 ) , . . . , N ( v k ) ar e dep endent. Pr o of. Since tr anslation and scaling do es not aﬀect dep endence, w.l.o.g. w e assume that v 0 = 0 and that the distance of the h yp erplane H fr om zero is one. Let h b e the unit v ector orthogonal to H . F or all i ∈ [ k ] we h av e N ( v i ) = v i / h v i , h i . Assume that v 0 , v 1 , . . . , v k are dep end en t, that is, w.l.o.g. v k = P i ∈ [ k − 1] a i v i for some a 1 , . . . , a k − 1 . F or all i ∈ [ k − 1] deﬁne b i = a i h v i , h i / h v k , h i . Th us N ( v k ) = P i ∈ [ k − 1] a i v i / h v k , h i = P i ∈ [ k − 1] b i N ( v i ) where P i ∈ [ k − 1] b i = 1, w hic h means that N ( v 1 ) , . . . , N ( v k ) are dep enden t. Since the map a i 7→ b i is inv ertible, the other d irection of the claim holds as w ell. W e ﬁrs t pr o v e the theorem for δ -SG ∗ k conﬁgurations. Pr o of of The or em 5.3. The pro of is b y indu ction on k . F or k = 1 w e kno w f ( δ , 1) ≤ cδ − 2 with c > 1 a universal constan t. Sup p ose k > 1. W e separate in to t wo cases. The ﬁrst case is when 16 V is an ( δ / (2 k ))-SG 1 conﬁguration and we are d on e u sing the b ou n d on k = 1. In the other case th er e is some p oint v 0 ∈ V s.t. the size of the set of p oin ts on sp ecial lines through v 0 is at most δ / (2 k ) (a line is sp ecial if it conta ins at least three p oin ts). Let S denote the set of p oin ts on sp ecial lines through v 0 . Thus | S | < δ n/ (2 k ). Let N : C d 7→ C d b e a normalization w .r .t. v 0 . Notice that for p oin ts v 6∈ S the image N ( v ) determines v . Similarly , all p oin ts on some sp ecial line map to the same p oin t via N . Our goal is to show that V ′ = N ( V \ { v 0 } ) is a ((1 − 1 / (2 k )) δ )- S G ∗ k − 1 conﬁguration (after eliminating multiplicitie s from V ′ ). This will complete th e pr o of since d im( V ) ≤ dim( V ′ ) + 1. Indeed, if this is th e case we ha ve f ( δ , k ) ≤ max { 4 c ( k/δ ) 2 , f ((1 − 1 / (2 k )) δ , k − 1) + 1 } . and by in d uction we h a v e f ( δ , k ) ≤ 4 c ( k /δ ) 2 . Fix v ′ 1 , . . . , v ′ k − 1 ∈ V ′ to b e k − 1 indep en d en t p oints (if no such tuple exists then V ′ is tr ivially a conﬁ gu r ation). Let v 1 , . . . , v k − 1 ∈ V b e p oin ts s.t. N ( v i ) = v ′ i for i ∈ [ k − 1]. Claim 5.5 implies that v 0 , v 1 , . . . , v k − 1 are in dep end en t. Th us, there is a set U ⊂ V of size at least δ n s.t. for ev ery u ∈ U either u ∈ ﬂ( v 0 , v 1 , . . . , v k − 1 ) or the k -ﬂat ﬂ( v 0 , v 1 , . . . , v k − 1 , u ) con tains a p oin t w outside ﬂ( v 0 , v 1 , . . . , v k − 1 ) ∪ { u } . Let ˜ U = U \ S so that N is inv ertible on ˜ U and | ˜ U | ≥ | U | − | S | ≥ (1 − 1 / (2 k )) δ n. Supp ose u ∈ ˜ U and let u ′ = N ( u ). B y Claim 5.5 if u ∈ ﬂ( v 0 , v 1 , . . . , v k − 1 ) then u ′ is in ﬂ( v ′ 1 , . . . , v ′ k − 1 ). Otherw ise, ﬂ( v 0 , v 1 , . . . , v k − 1 , u ) con tains a p oint w outside ﬂ( v 0 , v 1 , . . . , v k − 1 ) ∪ { u } . Let w ′ = N ( w ). W e will sh o w that w ′ is (a) conta ined in the ( k − 1)-ﬂat ﬂ( v ′ 1 , . . . , v ′ k − 1 , u ′ ) and (b) is outsid e ﬂ( v ′ 1 , . . . , v ′ k − 1 ) ∪{ u ′ } . Pr op ert y (a) follo ws from Claim 5.5 s ince v 0 , v 1 , . . . , v k − 1 , u, w are dep enden t and so v ′ 1 , . . . , v ′ k − 1 , u ′ , w ′ are also dep end en t. T o sho w (b) observe ﬁrst that b y C laim 5.5 the p oin ts v ′ 1 , . . . , v ′ k − 1 , u ′ are in d ep end en t (since v 0 , v 1 , . . . , v k − 1 , u are indep en- den t) and so u ′ is not in ﬂ( v ′ 1 , . . . , v ′ k − 1 ). W e also n eed to sho w that w ′ 6 = u ′ but this follo w s from the fact that u 6 = w and so w ′ = N ( w ) 6 = N ( u ) = u ′ since N is inv ertible on ˜ U and u ∈ ˜ U . Since | N ( ˜ U ) | = | ˜ U | ≥ (1 − 1 / (2 k )) δ n ≥ (1 − 1 / (2 k )) δ | V ′ | the p r o of is complete. W e can now prov e the theorem for δ -SG k conﬁgurations. Pr o of of The or em 5.4. The pro of follo ws b y induction on k (the case k = 1 is giv en by Theo- rem 4.3). Supp ose k > 1. Supp ose that dim( V ) > g ( δ, k ). W e wan t to sh o w that there exist k indep en d en t p oints v 1 , . . . , v k s.t. for at least 1 − δ fraction of the p oin ts w ∈ V w e h a v e that w is not in ﬂ( v 1 , . . . , v k ) and the ﬂat ﬂ( v 1 , . . . , v k , w ) is elementa ry (i.e. do es not conta in an y other p oin t). Let k ′ = g (1 , k − 1). By choice of g w e h a ve g ( δ, k ) > f ( δ, k ′ + 1) with f from Theorem 5.3. Th us, by Theorem 5.3, we can ﬁnd k ′ + 1 ind ep endent p oin ts v 1 , . . . , v k ′ +1 s.t. there is a set U ⊂ V of size at least (1 − δ ) n s.t. for ev ery u ∈ U we ha v e that u is n ot in ﬂ( v 1 , . . . , v k ′ +1 ) and the ( k ′ + 1)-ﬂat ﬂ ( v 1 , . . . , v k ′ +1 , u ) conta ins only one p oint, namely u , outside ﬂ ( v 1 , . . . , v k ′ +1 ). 17 W e no w apply the in ductiv e hypothesis on the set V ∩ ﬂ( v 1 , . . . , v k ′ +1 ) whic h has d imension at least k ′ = g (1 , k − 1). T his giv es us k indep enden t p oin ts v ′ 1 , . . . , v ′ k that deﬁne an elemen tary ( k − 1)-ﬂat ﬂ( v ′ 1 , . . . , v ′ k ). (Saying th at V is not 1-SG k − 1 is the same as sa yin g that it cont ains an elemen tary ( k − 1)-ﬂat). J oining an y of the p oin ts u ∈ U to v ′ 1 , . . . , v ′ k giv es us an elemen tary k -ﬂat and so the theorem is pr o v ed. 6 Generalizations of the Motzkin-Rabin T h eorem In this section w e pr o v e t wo v ariants of the Motzkin-Rabin Th eorem. The ﬁrst is a quantit ativ e analog in th e spirit of Theorem 4.3. T h e second is a v arian t in whic h the num b er of colors is three (instead of t w o). 6.1 A Quan titative V ariant Deﬁnition 6.1 ( δ -MR conﬁguration) . Let V 1 , V 2 b e t w o disjoin t ﬁnite su bsets of C d . Poin ts in V 1 are of c olor 1 and p oin ts in V 2 are of c olor 2. A line is call ed bi-chr omatic if it co nta ins at least one p oin t from eac h of the t wo colors. W e sa y that V 1 , V 2 are a δ -MR conﬁguration if for ev ery i ∈ [2] and f or ev er y p oint p ∈ V i , the bi-chromatic lines through p contai n at least δ | V i | p oints. Theorem 6.2. L et V 1 , V 2 ⊂ C d b e a δ -MR c onﬁgur ation. Then dim ( V 1 , V 2 ) ≤ O (1 /δ 4 ) . Pr o of. W e will ca ll a lin e passing through exactly tw o p oint s in V 1 (resp. V 2 ) a V 1 -ordinary (resp. V 2 -ordinary) line. W.l.o.g . assum e | V 1 | ≤ | V 2 | . W e sep erate the pro of into t wo cases: Case I is wh en V 2 is a ( δ / 2)-SG conﬁguration. Then, by Theorem 4.3, d im( V 2 ) ≤ O (1 /δ 2 ). If in addition dim( V 1 ) ≤ 13 / ( δ / 2) 2 then w e are done. Otherwise, b y Theorem 4.3, there exists a p oint a 0 ∈ V 1 suc h that th ere are at least (1 − δ / 2) | V 1 | V 1 -ordinary lines through a 0 . Let a 1 , . . . , a k denote the p oin ts in V 1 that b elong to these lines with k ≥ (1 − δ / 2) | V 1 | . W e now claim that V 2 ∪ { a 0 } spans all the p oin ts in V 1 . This will suﬃce sin ce, in this case, dim( V 2 ) ≤ O (1 /δ 2 ). Let a ∈ V 1 . Then, since V 1 , V 2 is a δ -MR conﬁguration, there are at least δ | V 1 | p oin ts in V 1 suc h that th e line through them and a conta ins a p oin t in V 2 . One of these p oints m ust b e among a 1 , . . . , a k , say it is a 1 . Since a is in the span of V 2 and a 1 and since a 1 is in the span of V 2 and a 0 w e are done. Case I I is wh en V 2 is not a ( δ / 2)-SG conﬁguration. In this case, there is a p oint b ∈ V 2 suc h that there are at least (1 − δ / 2) | V 2 | V 2 -ordinary lines through b . F r om th is fact and f rom the δ -MR p r op erty , we get th at | V 1 | ≥ ( δ / 2) | V 2 | (there are at least ( δ / 2) | V 2 | V 2 -ordinary lines through b that ha ve an additional p oin t from V 1 on th em). This implies that the union V 1 ∪ V 2 is a ( δ 2 / 4)-SG conﬁgur ation and the result follo ws b y app lying Theorem 4.3. 18 6.2 A Three Colors V arian t Deﬁnition 6.3 (3MR conﬁguration) . Let V 1 , V 2 , V 3 b e th r ee pairwise d isj oin t ﬁnite su bsets of C d , eac h of distin ct p oin ts. W e sa y that V 1 , V 2 , V 3 is a 3MR -conﬁguration if every line ℓ so that ℓ ∩ ( V 1 ∪ V 2 ∪ V 3 ) has more than one p oin t intersect s at least t w o of the sets V 1 , V 2 , V 3 . Theorem 6.4. L et V 1 , V 2 , V 3 b e a 3MR c onﬁgur ation and denote V = V 1 ∪ V 2 ∪ V 3 . Then dim ( V ) ≤ O (1) . Pr o of. Assume w.l.o.g. that V 1 is not smaller than V 2 , V 3 . Let α = 1 / 16 . There are sev eral cases to consider: 1. V 1 is an α -SG conﬁguration. By Theorem 4.3, th e dimension of V 1 is at most d 1 = O (1 /α 2 ) . Consider the t wo sets V ′ 2 = V 2 \ sp an( V 1 ) and V ′ 3 = V 3 \ sp an( V 1 ) , eac h is a set of distinct p oin ts in C d . Assu me w.l.o.g. that | V ′ 2 | ≥ | V ′ 3 | . 1.1. V ′ 2 is an α -SG conﬁguration. By Th eorem 4.3, the dimension of V ′ 2 is at most d 2 = O (1 /α 2 ) . Fix a p oin t v 3 in V ′ 3 . F or ev ery p oint v 6 = v 3 in V ′ 3 the line through v 3 , v cont ains a p oint f rom span( V 1 ) ∪ V ′ 2 . Th erefore, dim( V ) ≤ d 1 + d 2 + 1 ≤ O (1) . 1.2. V ′ 2 is not an α -SG conﬁguration. There is a p oin t v 2 in V ′ 2 so that for k ≥ | V ′ 2 | / 2 of th e p oin ts v 6 = v 2 in V ′ 2 the lin e th rough v 2 , v do es not con tain an y other p oin t from V ′ 2 . If V ′ 2 = sp an( V 1 , v 2 ) then the dimension of V 1 ∪ V 2 is at most d 1 + 1 and w e are done as in the previous case. Otherwise, there is a p oin t v ′ 2 in V ′ 2 \ sp an( V 1 , v 2 ). W e claim that in this case | V ′ 3 | ≥ k / 2. Denote by P 2 the k p oin ts v 6 = v 2 in V ′ 2 so th at the line through v 2 , v do es n ot con tain an y other p oin t from V ′ 2 . F or eve ry v ∈ P 2 there is a p oin t V 1 , 3 ( v ) in V 1 ∪ V 3 that is on the line through v , v 2 (the p oint v 2 is ﬁxed). There are t wo cases to consid er. The ﬁr st case is that f or at least k / 2 of the p oin ts v in P 2 w e h a v e V 1 , 3 ( v ) ∈ V 3 . In this case clearly | V 3 | ≥ k / 2. The seco nd case is that for at least k / 2 of the p oint s v in P 2 w e h a v e V 1 , 3 ( v ) ∈ V 1 . Fix suc h a p oint v ∈ P 2 (whic h is in span( V 1 , v 2 )). Th e line th rough v ′ 2 , v con tains a p oint v ′ from V 1 ∪ V 3 . T h e p oin t v ′ is n ot in span( V 1 ), as if it wa s then v ′ 2 w ould b e in span( v , v ′ ) ⊆ span( V 1 , v ). Therefore v ′ is in V 3 . Th is also imp lies that | V ′ 3 | ≥ k/ 2. 19 Denote V ′ = V 2 ∪ V ′ 3 . So we can conclude that for every v ′ in V ′ the sp ecial lines through v ′ con tain at least | V ′ | / 8 of the p oints in V 1 ∪ V 2 ∪ V 3 . As in the pro of of Theorem 4.3, we can th us deﬁ ne a family of triples T , eac h triple of th r ee distinct collinear p oints in V , so that eac h v ′ in V ′ b elongs to at least | V ′ | / 8 trip les in T and eac h tw o distinct v ′ , v ′′ in V ′ b elong to at most 6 tr iples. By a slight abuse of notation, w e also denote b y V the matrix with ro ws deﬁned b y the p oints in V . Let V 1 b e th e submatrix of V with ro w deﬁned b y p oin ts in span( V 1 ) ∩ V and V ′ b e the submatrix of V with ro w deﬁned b y p oin ts in V ′ . Use the triples in T to construct a matrix A so that A · V = 0. Let A 1 b e the submatrix of A consisting of th e columns that corresp ond to span( V 1 ) ∩ V and A ′ b e the submatrix of A consisting of the columns that corresp on d to V ′ . Th erefore, A ′ · V ′ = − A 1 · V 1 whic h imp lies rank( A ′ · V ′ ) ≤ rank( A 1 · V 1 ) ≤ d 1 . By th e ab o v e discussion A ′ is a (3 , | V ′ | / 8 , 6)-design matrix and th u s, b y Theorem 3.2, has rank at least | V ′ | − O (1) and so dim( V ′ ) ≤ O (1) + d 1 ≤ O (1) . W e can ﬁnally conclude that dim( V ) ≤ d 1 + dim ( V ′ ) ≤ O (1) . 2. V 1 is not an α -SG conﬁguration. There is a p oin t v 1 in V 1 so that for at least | V 1 | / 2 of the p oin ts v 6 = v 1 in V 1 the line through v 1 , v do es not con tain any other p oin t f rom V 1 . Assume w.l.o.g. that | V 2 | ≥ | V 3 | . This implies that | V 2 | ≥ | V 1 | / 4 . 2.1. | V 3 | < | V 2 | / 16 . In this case the conﬁguration d eﬁned by V 1 ∪ V 2 is an α -SG conﬁgu- ration. By T heorem 4.3, the dimension of V 1 ∪ V 2 is at most d 1 , 2 = O (1 /α 2 ) . Fix a p oin t v 3 in V 3 . F or every p oin t v 6 = v 3 in V 3 the line through v 3 , v cont ains a p oint f rom V 1 ∪ V 2 . Therefore, dim( V ) ≤ d 1 , 2 + 1 ≤ O (1) . 2.1. | V 3 | ≥ | V 2 | / 16 . In th is case V is an α -SG conﬁguration. By Th eorem 4.3, the dimen- sion of V is thus at most O (1 /α 2 ). 20 7 Tw o-Qu ery Lo cally Correctable Co des W e n o w prov e the non-existence of 2-query (linear) lo cally correctable co des (LCC ) o v er C . W e start by form ally deﬁn ing lo cally correctable codes: Deﬁnition 7.1 (Linear locally co rrectable code (LCC)) . Let F b e some ﬁeld. A ( q , δ )-LCC o ver F is a linear sub space C ⊂ F m suc h that there exists a randomized deco ding pro cedu re D : F m × [ m ] 7→ F with the follo wing prop erties: 1. F or all x ∈ C , for all i ∈ [ m ] and for all v ∈ F m with w ( v ) ≤ δ m w e ha v e that D ( x + v , i ) = x i with p robabilit y at least 3 / 4 (the probabilit y is tak en only o ve r the in ternal randomness of D ). 2. F or every y ∈ F m and i ∈ [ m ], th e deco der D ( y , i ) r eads at most q p ositions in y . The dimension of an LCC is simply its dimension as a subs pace of F m . In the abov e d eﬁnition we allo w the algorithm D to p erform op er ations o v er the ﬁeld F . Since we do not care ab out the run ning time of D we d o not discuss issues of represen tation of ﬁeld ele ments and eﬃciency of handling them. (In an y case, it turn s out that for linear co d es in the small n umb er of qu eries and lo w error case, one can assume w.l.o.g. that the deco der is also linear, s ee Lemma 7.4 b elo w.) Our result on lo cally deco dable co des is the follo wing: Theorem 7.2 (Restatemen t of Theorem 3— non-existence of 2 query LCCs o ver C ) . L et C ⊂ C m b e a (2 , δ ) -LCC over C . Then dim( C ) ≤ O (1 /δ 9 ) . As in T heorem 4.3, also in this theorem, δ can b e an arbitrary function of m . T o make the connection b et ween LC C s an d S G -conﬁgurations explicit, we deﬁne the notion of a δ -LCC conﬁguration. Deﬁnition 7.3 ( δ -LCC Con ﬁ guration) . A list of non-zero p oints ( v 1 , . . . , v m ) in C d (not neces- sarily d istinct) is called a δ -LCC conﬁguration if for ev ery subset ∆ ⊂ [ m ] of size at most δ m and for ev ery i ∈ [ m ], there exist j, k ∈ [ m ] \ ∆ suc h that either v i ∈ { v j , v k } (in whic h ca se v i can b e reco ve red b y its o wn copies), or v i , v j , v k are three distinct colli near p oint s (in which case v i is reco v ered by t wo other co ord inates). The follo wing lemma sh o ws the connection b etw een these t wo notions. Lemma 7.4. If ther e exists a (2 , δ ) -LCC of dimension n over C then ther e exists a δ -LCC c onﬁgur ation of dimension at le ast n − 1 over C . T o prov e the lemma w e will u se the f ollo wing deﬁ n ition. Deﬁnition 7.5 (Generating s et) . L et C ⊂ F m b e a subsp ace. W e sa y that a list of ve ctors V = ( v 1 , . . . , v m ) in F n is a ge ner ating set for C if C = { ( h y , v 1 i , h y , v 2 i , . . . , h y , v m i ) | y ∈ F n } , 21 where h y , v i is the standard inner p ro duct o v er F . Pr o of of L emma 7.4. Let V = ( v 1 , . . . , v m ) b e a generating set for C with dim( V ) ≥ n − 1. W e m igh t lose 1 since we deﬁn ed dim( V ) as the dimen sion of the smallest aﬃne sub s pace con taining V . When the lo cal deco d er for C r eads t wo p ositions in a codeword, it is actually reading h y , v j i , h y , v k i for some v ector y ∈ C n (or n oisy v ersions of them). In ord er to b e able to reco ver h y , v i i fr om h y , v j i , h y , v k i with p ositive p r obabilit y it m ust b e that v i ∈ sp an { v j , v k } . (If w e c h o ose y as Gaussian and v i is not in th e span of v j , v k then ev en conditioned on the v alues of h y , v j i , h y , v k i the r .v. h y , v i i tak es an y sp eciﬁc v alue with pr obabilit y zero.) Applying an inv ertible linear transform ation on V preserves prop erties suc h as one v ector b eing in the span of another set. So we can assume w.l.o.g. that the ﬁrst co ordinate in all elements of V is non-zero. Scaling eac h v i b y a non-zero scalar also preserv es th e prop erties of spans and so we can assum e w.l.o.g. that the ﬁrst co ordinate in eac h v i is equ al to 1. Now, for v i to b e in the span of v j , v k it m ust b e that either v i ∈ { v j , v k } or v i is on the line passing through v j , v k (and they are all distinct). Th us, we hav e a δ -LCC conﬁguration with dimension n − 1. In view of this lemma, in order to pro v e Theorem 7.2 it is enough to prov e: Theorem 7.6. L et V = ( v 1 , . . . , v m ) ∈ ( C d ) m b e a δ -LCC c onﬁgur ation. Then dim ( V ) ≤ O (1 /δ 9 ) . 7.1 Pro of of Theorem 7.6 Let V = ( v 1 , . . . , v m ) b e the list of m p oin ts in C d . T h e main diﬃculty in pr o v in g the theorem is that some of these p oin ts may b e the same. That is, t wo p oint s v i , v j can actually corresp ond to the same vecto r in C d . In this case w e sa y that v i , v j are c opies of eac h other. Otherw ise, w e sa y that v i , v j are distinct . If v is a p oin t in the list V , w e let th e multiplicity of v , denoted M ( v ), b e the n umb er of times that (a copy of ) v o ccurs in V . W e n ote th at wh ile rep etitio ns make the pr o of of Theorem 7.6 more complicated, w e do not kno w if th ey actually help in constructing LCCs with b etter parameters. Our pro of will pr o ceed in an iterativ e wa y , at eac h step identifying a suﬃcientl y large su blist with small d imension and remo ving it. The ke y step will b e the follo wing theorem: Theorem 7.7. Ther e exists an inte ger K 1 > 0 s.t. the fol lowing holds. L et V = ( v 1 , . . . , v m ) ∈ ( C d ) m b e a δ -LCC c onﬁgur ation. Then ther e exists a sublist V ′ ⊂ V of size at le ast δ 3 m/K 1 and dimension at most K 1 /δ 6 . Pr o of. If there exists a p oin t v ∈ V with multiplic it y large r that δ m / 10 then the theorem is true by taking V ′ to b e all copies of this p oint. This av oids the case where a p oin t is reco v ered mostly by its o wn copies. F or the rest of the pro of we can, thus, assume the follo wing. F act 7.8. F or al l v ∈ V and for every sublist ∆ of V of size at most δ m / 2 ther e is a c ol line ar triple c ontaining v such that the other two p oints in the triple ar e not in ∆ (and ar e distinct fr om v ). 22 W e will d escrib e a (probabilistic) constru ction of a family of collinear trip les and build a design matrix from it. W e call a trip le of p oin ts in V go o d if it con tains three distinct collinear p oints. W e d eﬁne a family T of goo d triples as follo ws: F or every line ℓ that has at least three distinct p oin ts in V w e will d eﬁne (randomly) a family T ℓ of go o d triples (la ter w e will ﬁ x the randomness). The family T will b e the union of all these s ets. Remark 7.9. The construction of T we present is p robabilistic. It is p ossib le to construct T explicitly and ac hiev e s im ilar pr op erties. W e c ho ose to present the probabilistic construction as it is sim p ler and less tec hnical. Let ℓ b e suc h a line w ith r p oints on it (coun ting multiplicit ies). Denote by V ( ℓ ) the sublist of V contai ning all p oints that lie on ℓ . W e ﬁrst tak e the family F of triples on [ r ] giv en by Lemma 4.6 and then pic k a random one-to-one mapp ing ρ : [ r ] 7→ V ( ℓ ). F or a triple t in F we denote by ρ ( t ) th e triple of p oin ts in V ( ℓ ) that is the image of t under ρ . W e tak e T ℓ to b e the set of all triples ρ ( t ) with t ∈ F and suc h that ρ ( t ) is goo d (i.e., it ‘hits’ thr ee distinct p oin ts). In tuitiv ely , we w ill ha ve man y go o d trip les on a line (i n expectation) if th er e are no t wo p oints whose copies co v er most of the line (then the pr obabilit y of hitting th r ee distinct p oin ts is small). W e will later sho w that this cannot h app en on too man y lines. The next prop ositio n shows that there is a wa y to ﬁx the randomness so that T con tains a quadratic num b er of trip les. Prop osition 7.10. The exp e ctation of | T | is at le ast αm 2 with α = ( δ / 15) 3 . W e will p ro v e this prop osition later in Section 7.2 and will conti nue no w with the pro of of the theorem. Fix T to b e a family of triples that has size at least th e exp ectation of | T | . By construction and Lemma 4. 6, the family T con tains only go o d triples and eac h pair of p oin ts app ears in at most 6 diﬀerent triples (since ev ery tw o distinct p oints deﬁne a s in gle line and t w o non-distinct p oints nev er app ear in a triple together). Th e family T thus deﬁn es a 3-regular hyp ergraph with v ertex set [ m ] and at least αm 2 edges and of co-degree at m ost 6. Lemma 4.9 thus implies that there is a sublist V ′ of V of size at least | V ′ | = m ′ ≥ αm/ 12 ≥ ( δ / 45 ) 3 m with the follo wing prop ert y: Let T ′ b e the subfamily of T that V ′ induces. Eve ry v ′ in V ′ is con tained in at least αm/ 2 triples in T ′ . By a slight abuse of notation, we also d en ote by V ′ the m ′ × d m atrix with rows d eﬁned by the p oint s in V ′ (including rep etitions). W e n o w use the tr iples in T ′ to construct a m atrix A ′ so that A ′ · V ′ = 0. By th e ab ov e discussion A ′ is a (3 , αm/ 2 , 6)-design matrix and thus, by Theorem 3.2, has rank at least m ′ −  18 m ′ αm  2 ≥ m ′ − (18 /α ) 2 and so dim( V ′ ) ≤ (18 /α ) 2 ≤ (60 /δ ) 6 as was requir ed. 23 The next pr op osition sho ws how the ab o ve th eorem can b e used r ep eatedly on a give n LCC. Prop osition 7.11. Ther e exist an inte ger K 2 > 0 s.t. the fol lowing holds: L et V = ( v 1 , . . . , v m ) ∈ ( C d ) m b e a δ -LCC c onﬁgur ation and let U, W b e a p artition of V into two disjoint sublists such that W ∩ sp an ( U ) = ∅ . Then ther e exists a new p artition of V to two sublists U ′ and W ′ such that W ′ ∩ sp an ( U ′ ) = ∅ and such that 1. | U ′ | ≥ | U | + δ 3 m/K 2 , and 2. dim ( U ′ ) ≤ dim ( U ) + K 2 /δ 6 . Pr o of. First, we can assume that all p oints in W ha ve multi plicit y at most δ m/ 2 (otherwise we can add one p oin t f r om W with high m ultiplicit y to U to get U ′ ). Thus, for all p oin ts v and all su blists ∆ of size at most δ m/ 2 th ere is a collinear tr iple of three distinct p oin ts conta ining v and t wo other p oin ts outside ∆. Again, this is to a void p oin ts that are reco v ered mostly by copies of themselv es. F or a p oint w ∈ W w e deﬁne three disjoint su blists of p oin ts U ( w ) , P 1 ( w ) and P 2 ( w ). Th e ﬁrst list, U ( w ), will b e the list of all p oints in U that are on sp ecial lines thr ough w (that is, lines con taining w and at least t w o other distinct p oin ts). No tice that, since w 6∈ span( U ), eac h line thr ough w can cont ain at most one p oin t fr om U . The second list, P 1 ( w ), w ill b e the list of p oints in W \ { w } that are on a line cont aining w and a p oint from U . T he thir d list, P 2 ( w ), will b e of all other p oin ts on sp ecial lines through w (that is, on sp ecial lines th at do not intersect U ). Th ese three lists are indeed disjoints, sin ce w is th e only common p oin t b et wee n t wo lines passing through it. By the ab o ve discussion w e h a v e that | P 1 ( w ) | + | P 2 ( w ) | ≥ δ m / 2 for all w ∈ W (sin ce removing these tw o lists destro ys all collinear triples with w ). W e no w separate the p r o of in to t wo cases: Case I : There exists w ∈ W with | P 1 ( w ) | > δ m/ 4 . In th is case we can simply tak e U ′ to b e the p oin ts in V that are also in the span of { w } ∪ U . This new U ′ will include all p oin ts in P 1 ( w ) and so will gro w b y at least δ m/ 4 p oints. Its dimension will gro w by at most one an d so w e are done. Case I I : F or all w ∈ W , | P 2 ( w ) | ≥ δ m/ 4 . Denote m ′ = | W | . In this case W itself is a δ ′ -LCC conﬁgur ation with δ ′ = δ m 8 m ′ . Applying Th eorem 7.7 we get a su blist U ′′ ⊂ W of size at least ( δ ′ ) 3 m ′ K 1 ≥ ( δ / 8) 3 · m K 1 and dimen s ion at most K 1 ( δ ′ ) 6 ≤ K 1 (8 /δ ) 6 . W e can thus tak e U ′ to b e the p oin ts in V that are in the span of U ∪ U ′′ and the p rop osition is prov ed. 24 Pr o of of The or em 7.6. W e apply Prop osition 7.1 1 on V , starting with the partition U = ∅ , W = V and ending when U = V , W = ∅ . W e ca n apply the prop osition at most K 2 /δ 3 times and in eac h step add at most K 2 /δ 6 to the dimension of A (whic h is initially zero). Therefore, the ﬁnal list U = V will h a v e dimension at most O (1 /δ 9 ). 7.2 Pro of of Prop osition 7.10 Order the p oin ts in V so that all copies of the same p oint are consecutiv e and so that M ( v i ) ≤ M ( v j ) wh enev er i ≤ j . Let S ⊂ V b e the sub list con taining the ﬁ rst δ m / 10 p oints in this ordering (w e may b e splitting the copies of a single p oint in the midd le but this is ﬁne). W e will use the follo wing simple fact later on: F act 7.12. If v ∈ S and M ( v ′ ) < M ( v ) then v ′ ∈ S . F or a p oin t v ∈ V w e denote b y T ( v ) the set of (ordered) triples in T con taining v and for a line ℓ by T ℓ ( v ) the set of (ord er ed ) triples in T ℓ con taining v . Recall that these are all r an d om v ariables d etermined by the c hoice of the mappin gs ρ for eac h line ℓ . The pr op osition will follo w b y the follo win g lemma. Lemma 7.13. L et v ∈ S . Then the exp e ctation of | T ( v ) | i s at le ast ( δ/ 10) 2 m . The lemma completes the p ro of of th e prop osition: summing o ver all p oin ts in S we get E [ | T | ] ≥ E " (1 / 3) X v ∈ V | T ( v ) | # (each triple is counted at most three times) ≥ (1 / 3 ) X v ∈ S E [ | T ( v ) | ] ≥ (1 / 3 ) · ( δ m/ 10) · (( δ / 10) 2 m ) ≥ ( δ/ 15) 3 m 2 . Pr o of of L emma 7.13. Denote b y L ( v ) the set of all sp ecial lines through v . T o pro v e the lemma w e will iden tify a subfamily L ′ ( v ) of L ( v ) that contributes man y triples to T ( v ). T o do so, w e need the follo wing deﬁnitions. F or a set γ ⊂ C d denote by P ( γ ) the set of distinct p oints in V that are in γ . Denote M ( γ ) = P v ∈ P ( γ ) M ( v ). Denote b y P ( ¯ S ) the set of distinct p oints n ot in S . Deﬁnition 7.14 (De generate line) . Let ℓ ∈ L ( v ). W e say that ℓ ∈ L ( v ) is de ge ner ate if either 1. Th e size of P ( ℓ ) ∩ P ( ¯ S ) is at m ost one. That is, ℓ con tains at most one distinct p oint outside S . Or, 2. Th er e exists a p oin t v ℓ ∈ P ( ℓ ), distinct fr om v , suc h that M ( v ℓ ) ≥ (1 − δ / 10) M ( ℓ ). A d egenerate line satisfying the ﬁrst (second) pr op ert y ab ov e will b e called a degenerate line of the ﬁ r st (second) kind. 25 Deﬁne L ′ ( v ) as the set of line ℓ in L ( v ) that are not degenerate. W e will con tin ue by pro ving t w o claims. The ﬁ rst claim s ho ws that ev ery line in L ′ ( v ) con tributes man y triples in exp ecta tion to T ( v ). Claim 7.15. F or eve ry ℓ ∈ L ′ ( v ) we have E [ | T ℓ ( v ) | ] ≥ δ M ( ℓ ) / 1 0 . Pr o of. Denote r = | M ( ℓ ) | . The family of triples T ℓ is obtained by taking a family of r ( r − 1) triples F on [ r ] (obtained from Lemma 4.6) and mapping it randomly to ℓ , omitting all triples that are not go o d (those that do not ha ve three d istinct p oin ts). F or eac h triple t ∈ F the probabilit y that ρ ( t ) w ill b e in T ℓ ( v ) can b e lo wer b ounded by 3 r · 2 r 3( r − 1) · δ 20 = δ 10( r − 1) The factor of 3 /r comes from the probability that one of the three en tries in t maps to v (these are disjoin t even ts so we can sum their p robabilities). The n ext factor, 2 r / (3( r − 1)) , comes from the probabilit y that the s econd entry in t (in some ﬁxed order) maps to a p oin t distinct from v . Ind eed s in ce | P ( ℓ ) ∩ P ( ¯ S ) | ≥ 2 and using F act 7.12 w e kn ow that there are at least t w o distinct p oin ts v ′ , v ′′ on ℓ with M ( v ′ ) ≥ M ( v ) and M ( v ′′ ) ≥ v . S ince M ( v ) + M ( v ′ ) + M ( v ′′ ) ≤ r , we get that M ( v ) ≤ r / 3, and so there are at least 2 r / 3 “goo d ” places for the second p oint to map to. The last factor, δ / 20, comes from the probabilit y th at the third element of the triple will map to a p oin t distinct from the ﬁr s t t wo. The b ound of δ / 20 will follo w from the f act that ℓ do es n ot satisfy the second prop erty in the deﬁnition of a d egenerate line. T o see why , let v 2 b e the image of the second entry in t . Since ℓ is not degenerate, r ′ , r − M ( v 2 ) > δ r / 10. S ince | P ( ℓ ) ∩ P ( ¯ S ) | ≥ 2, ther e is a p oin t v ′ in P ( ¯ S ) not in { v , v 2 } , and h ence, b y F act 7.12, M ( v ) ≤ M ( v ′ ). Sin ce M ( v ) + M ( v ′ ) ≤ r ′ , w e get that M ( v ) ≤ r ′ / 2. Thus r ′ − M ( v ) ≥ r ′ / 2 ≥ δ r / 20. But r ′ − M ( v ) is exactly the num b er of ‘go o d ’ places that the th ird entry can map to that are from v and v 2 . Using linearit y of exp ectat ion we can conclud e E [ | T ℓ ( v ) | ] ≥ r ( r − 1) · δ 10( r − 1) = δ r / 10 . The second claim shows that there are many p oin ts on lines in L ′ ( v ). Claim 7.16. With the ab ove notations, we have: X ℓ ∈ L ′ ( v ) M ( ℓ ) ≥ δ m/ 10 . Pr o of. Assume in con tradiction that X ℓ ∈ L ′ ( v ) M ( ℓ ) < δ m/ 10 . 26 Let ∆ ′ denote the su b-list of V con taining all p oints that lie on lines in L ′ ( v ) so that | ∆ ′ | ≤ δ m/ 10. W e w ill derive a con tradiction b y ﬁnd ing a small su blist ∆ of V (cont aining ∆ ′ and tw o other small s ub-lists) that w ould violate F act 7.8. Th at is, if we remov e ∆ from V , we destroy all collinear triples cont aining v . Let ℓ b e a d egenerate line of the s econd kind. Then there is a p oin t v ℓ on it th at is distinct from v and has m ultiplicit y at least (1 − δ / 10) M ( ℓ ). F or ev ery such line let ∆ ℓ denote th e sub list of V conta ining all of the at most ( δ / 10) M ( ℓ ) − M ( v ) p oints on this line that are d istinct from b oth v and v ℓ . Let ∆ 2 denote the union of th ese lists ∆ ℓ o ver all d egenerate lin es of the second kind. W e no w ha ve that | ∆ 2 | ≤ δ m/ 10 since P ℓ ( M ( ℓ ) − M ( v )) ≤ m and in eac h line ℓ we h a ve | ∆ ℓ | ≤ ( δ / 10) M ( ℓ ) − M ( v ) ≤ ( δ/ 10)( M ( ℓ ) − M ( v )) . Notice th at, remo vin g the p oin ts in ∆ 2 destro ys all collinear triples on degenerate lines of the second kind. Finally , let ∆ S denote the sub list of V conta ining all p oints that ha ve a cop y in S . Thus ∆ S con tains th e list S (of at most δ m/ 10 element s), plus all of the at most δ m/ 10 co pies of the last p oin t in S , meaning that | ∆ S | ≤ δ m/ 5. Remo ving ∆ S destro ys all collinear triples on degenerate lines of th e ﬁ rst kind. Deﬁne ∆ as th e union of the three sublists ∆ ′ , ∆ 2 and ∆ S . F rom the ab o ve w e h a v e th at remo ving ∆ from V destro ys all collinear trip les con taining V and that | ∆ | ≤ 4( δ / 10) m < δm/ 2. This con tradicts F act 7.8. Com bining the t w o claims we get that for all v ∈ S , E [ | T ( v ) | ] ≥ X ℓ ∈ L ′ ( v ) E [ | T ℓ ( v ) | ] ≥ X ℓ ∈ L ′ ( v ) δ M ( ℓ ) / 10 ≥ ( δ/ 10) · ( δ m/ 10) = ( δ / 10 ) 2 m. This completes the pro of of Lemma 7.13. 8 Extensions to Other Fields In this section we sho w that our r esults can b e extended from the complex ﬁeld to ﬁelds of c h aracteristic zero, and eve n to ﬁelds with v ery large p ositiv e charac teristic. The argumen t is quite generic and relies on Hilb ert’s Nullstellensatz . Deﬁnition 8.1 ( T -matrix) . Let m , n b e integ ers and let T ⊂ [ m ] × [ n ]. W e call an m × n matrix A a T -matrix if all entries of A with indices in T are non-zero and all entries with indices outside T are zero. Theorem 8.2 (Eﬀectiv e Hilbert’s Nullstellensatz [Kol88 ]) . L et g 1 , . . . , g s ∈ Z [ y 1 , . . . , y t ] b e de- gr e e d p olynomials with c o e ﬃcients i n { 0 , 1 } and let Z , { y ∈ C t | g i ( y ) = 0 ∀ i ∈ [ s ] } . Supp ose h ∈ Z [ z 1 , . . . , z t ] is ano ther p olynomial with c o e ﬃcients i n { 0 , 1 } which vanishes on Z . Then ther e exist p ositive inte gers p, q and p olynomials f 1 , . . . , f s ∈ Z [ y 1 , . . . , y t ] such that s X i =1 f i · g i ≡ p · h q . 27 F urthermor e, one c an b ound p and the maximal absolute value of the c o eﬃcients of the f i ’s by an explicit function H 0 ( d, t, s ) . Theorem 8.3. L et m , n, r b e inte gers and let T ⊂ [ m ] × [ n ] . Supp ose that al l c omplex T - matric es have r ank at le ast r . L et F b e a ﬁeld of either char acteristic zer o or of ﬁnite la r ge enough char acteristic p > P 0 ( n, m ) , wher e P 0 is some explicit function of n and m . Then, the r ank of al l T -matric es over F i s at le ast r . Pr o of. Let g 1 , . . . , g s ∈ C [ { x ij | i ∈ [ m ] , j ∈ [ n ] } ] b e the determinan ts of all r × r sub-matrices of an m × n matrix of v ariables X = ( x ij ). Th e statemen t “all T -matrices h av e rank at least r ” can b e phr ased as “if x ij = 0 for all ( i, j ) 6∈ T and g k ( X ) = 0 for all k ∈ [ s ] then Q ( i,j ) ∈ T x ij = 0.” That is, if all ent ries outside T are zero and X has ran k smaller than r then it must hav e at least one zero entry also inside T . F rom Nullstellensatz we know that there are in tegers α, λ > 0 and p olynomials f 1 , . . . , f s and h ij , ( i, j ) 6∈ T , with integ er coeﬃcients such th at α ·   Y ( i,j ) ∈ T x ij   λ ≡ X ( i,j ) 6∈ T x ij · h ij ( X ) + s X k =1 f i ( X ) · g i ( X ) . (2) This id entit y implies the high rank of T -matrices also ov er any ﬁeld F in wh ic h α 6 = 0. Since w e ha v e a b oun d on α in terms of n and m the result follo ws. 9 Discussion and O p en Problems Our rank b oun d f or design matrices has a d ep enden ce on q , the num b er of non-zeros in eac h ro w . Can this dep endency b e remov ed? This migh t b e p ossible s in ce a b oun d on q follo ws ind irectly from sp ecifying the b ound on t , th e sizes of the intersectio ns. Remo ving this dep endency migh t also enable u s to argue ab ou t squ are matrices. Our results so far are interesting only in the range of p arameters wh ere the num b er of rows is m uch larger than the n umber of columns. With resp ect to Sylv ester-Gallai conﬁgurations, th e most ob v ious op en problem (d iscussed in the intro d uction) is to close the gap b etw een our b ound of O (1 /δ 2 ) on the dimension of δ -S G conﬁguration and the trivial lo we r b ound of Ω(1 /δ ) obtained b y a simple partition of the p oin ts in to 1 /δ lines. Another in teresting direction is to explore fu rther the connection b etw een d esign-matrices and LCCs. The most natural wa y to construct an LCC is by starting with a lo w-rank d esign matrix and then deﬁning the co de b y taking the matrix to b e its parit y-c hec k matrix. Call suc h co des design-LCCs . Our result on th e r ank of design matrices sho ws, essenti ally , that design- LCCs ov er the complex num b ers cann ot hav e go o d p arameters in general (ev en for large qu er y complexit y). It is natural to ask whether there could exist LC Cs th at d o not originate from designs. Or, more sp eciﬁcally , whether any LCC deﬁnes another LCC (w ith similar parameters) whic h is a design-LCC. T his qu estion w as already raised in [BIW07]. Answering th is qu estion o ver th e complex n umb ers will, using our results, giv e b oun ds for general LCC s . It is not out of the question to hop e for b oun d s on LCCs with qu er y complexit y as large as p olynomial in m (the enco ding length). This wo uld b e enough to deriv e new results on r igidit y via the connection 28 made in [Dvi10 ]. In particular, our results on design matrices still giv e meaningful b ounds (on design-LCCs) in this range of parameters. More form ally , our results suggest a b ound of roughly p oly( q , 1 /δ ) on the d imension of ( q , δ )- LCCs that arise fr om designs. A strong from of a conjecture from [Dvi10] s a y s that an LCC C ⊂ F n with q = n ǫ queries and error δ = n − ǫ , for some constant ǫ > 0, cannot h av e dimension 0 . 99 · n . This conjecture, if true, w ould lead to n ew results on rigidit y . Thus, sho wing that any LCC deﬁnes a d esign (up to some p olynomial loss of p arameters), com bined with our results, w ould lead to new r esults on r igidit y . Ac kno wledgemen ts W e thank Moritz Hardt for many helpfu l con versatio ns. W e thank Jozsef Solymosi for h elpful commen ts. References [Alo09] Noga Alon. P erturb ed identit y matrices h a v e h igh rank: P r o of and applications. Comb. Pr ob ab. Comput. , 18(1-2 ):3–15, 20 09. [Bar98] F ran ck Barthe. On a rev er s e form of th e brascamp-lieb inequalit y . Inventiones Math- ematic ae , 134:335– 361, 1998 . 10.1007 /s002220050 267. [BE67] W. 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Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

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