QUASIRANDOMNESS AND GOWERS THEOREM. August 16, 2007

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1 QUASIRANDOMNESS AND GOWERS THEOREM QIAN ZHANG August 16, 2007 Abstract. Quasiradomess will be described ad the Quasiradomess Theorem will be used to prove Gowers Theorem. This article assumes some familiarity with liear algebra ad elemetary probability theory. Cotets 1. Lidsey s Lemma: A Illustratio of quasiradomess How Lidsey s Lemma is a Quasiradomess result 2 2. The Quasiradomess Theorem How the Quasiradomess Theorem is a quasiradomess result Gowers theorem Traslatig Gowers Theorem: Provig m Proves Gowers Theorem Provig m 11 Refereces Lidsey s Lemma: A Illustratio of quasiradomess Defiitio 1.1. A is a Hadamard matrix of size if it is a matrix with each etry (a ij ) either +1 or -1. Moreover, its rows are orthogoal, i.e. ay two distict row vectors have ier product 0. Remark 1.2. If A is a matrix with orthogoal rows, the AA T = I, so ( 1 A)( 1 A) T = I ( 1 A) T = ( 1 A) 1 A T A = I, so the colums of A are also orthogoal, ad ( 1 A) T ( 1 A) = I. Notatio deotes the colum vector that has 1 as every compoet. Remark 1.4. For ay matrix A, 1 T A 1 is the sum of the etries of A. 1 Fact 1.5. x 2 = x T x. Hece, A x 2 = (A x) T (A x) = x T A T A x. Defiitio 1.6. Give a matrix A ad a submatrix T of A, let X be the set of rows of T ad let Y be the set of colums of T. x is a icidece vector of X if for all compoets x i of x, x i = 1 if the ith row vector of A is a row vector of T 1 1 selects colums of A ad sums their correspodig compoets. 1 T selects rows of A, selectig ad addig together certai compoet sums. Replacig the ith compoet of 1 with a 0 would deselect the ith colum of A ad replacig the ith compoet of 1 T with 0 would deselect the ith row of A. 1

2 2 QIAN ZHANG ad x i = 0 otherwise. y is a icidece vector of Y if for all compoets y i of y, y i = 1 if the ith colum vector of A is a colum vector of T ad y i = 0 otherwise. Lemma 1.7. (Lidsey s Lemma) If A = (a ij ) is a Hadamard matrix ad T is a k l-submatrix, the (i,j) T a ij kl Proof. Let X be the set of rows of T ad let Y be the set of colums of T. Note that X = k ad Y = l. Let x be the icidece vector of X ad let y be the icidece vector of Y. The sum of all etries of T is (i,j) T a ij, which is x T A y (recall 1.4), so x T A y = (i,j) T a ij. By the Cauchy-Schwarz iequality, x T A y x A y. By 1.2 ad 1.5, 1 A y = y T ( 1 A) T ( 1 A) y = y T y = y A y = ( 1 A y) = 1 A y = y Substitutig for A y i the Cauchy-Schwarz iequality ad otig that x = k ad y = l, x T A y x ( y ) = kl How Lidsey s Lemma is a Quasiradomess result. The followig corollary illustrates how Lidsey s Lemma is a quasiradomess result. It says that if T is a sufficietly large submatrix, the the umber of +1 s ad the umber of -1 s i T are about equal. Corollary 1.8. Let T be a k l submatrix of a Hadamard matrix A. If kl 100, the the umber of +1 s ad the umber of -1 s each occupy at least 45% ad at most 55% of the cells of T. Proof. Let x be the umber of +1 s i T ad let y be the umber of -1 s i T. Suppose kl 100. We wat to show that (0.45)kl x (0.55)kl ad (0.45)kl y (0.55)kl. By Lidsey s Lemma, (i,j) T a ij kl. Note that x y = (i,j) T a ij, so (i,j) T a ij = x y kl. We kow that k > 0 ad l > 0, so kl > 0. x y kl kl (kl) 2 = kl 100 = 1 10 where the last iequality holds because kl 100. Sice all etries of T are either +1 or -1, the sum of the umber of +1 s ad the umber of -1 s is the umber of etries i T, so x + y = kl, hece y = kl x. Substitutig i for y, x (kl x) kl kl 10 9kl 20 9kl 20 9kl x kl kl 10 x 11kl 20 kl x 11kl 20 y 11kl 20

3 QUASIRANDOMNESS AND GOWERS THEOREM 3 Defiitio 1.9. A radom matrix is a matrix whose etries are radomly assiged values. Etries assigmets are idepedet of each other. To see how Corollary 1.8 shows Hadamard matrix A to be like a radom matrix but ot a radom matrix, cosider a radom matrix B whose etries are assiged either +1 or -1 with probability p ad 1 p respectively. Cosider U, a k l submatrix of B. U has kl etries, ad w, the umber of etries of the kl etries that are +1, would be a radom variable. Cosiderig U s etries assigmets idepedet trials that result i either success or failure ad callig the occurrece of +1 a success, P (w = s) is the probability of s successes i kl idepedet trials, which is the product of the probability of a particular sequece of s successes, p s (1 p) kl s, ad the umber of such se- ( ) ( ) queces, kl s, so P (w = s) = kl s p s (1 p) kl s. I other words, w has a biomial probability distributio. Hece, w has expected value klp. If each etry has equal probability of beig assiged +1 or -1, p = 1 2 so E(w) = kl( 1 2 ). Note that w ca take values far from E(w), sice P (w = s) shows w has ozero probability of beig ay iteger s where 0 s kl. Now cosider Hadamard matrix A, its k l submatrix T, ad x, the umber of +1 s i T. Corollary 1.8 shows that x must take values close to E(w). 2 More precisely, if kl 100, x must be withi 5% of E(w). x is like radom w i that we ca expect x to take values close to the expected value of w. However, x is ot radom because it must be withi 5% of E(w), while radom w ca take values farther from E(w), ay value ragig from 0 to kl. 3 The above argumet is symmetrical: It ca be used to compare y, the umber of -1 s i T, ad z, the umber of -1 s i U. I derivig P (w = s), we called the occurrece of +1 a success. We could have arbitrarily called the occurrece of -1 a success. The P (z = s) = ( kl s ) p s (1 p), E(z) = kl( 1 2 ) if p = 1 2, ad y would be like radom z, but ot radom, i the same way that x would be like radom w, but ot radom. I short, Hadamard matrix A is quasiradom because it is like a radom matrix B, but ot itself a radom matrix. Characteristics (x ad y) of k l T, a sufficietly large 4 submatrix of A, are similar to characteristics (w ad z) of k l U, a submatrix of B. A is like, but ot, a radom matrix B because submatrices of A have properties similar to, but ot the same as, submatrices of B. 2. The Quasiradomess Theorem Defiitio 2.1. A graph G = (V, E) is a pair of sets. Elemets of V are called vertices ad elemets of E are called edges. E cosists of distict, uordered pairs of vertices such that o vertex forms a edge with itself: v V, E V V \{v, v}. v 1, v 2 V are adjacet whe {v 1, v 2 } E, deoted v 1 v 2. The degree of a vertex is the umber of vertices with which it forms a edge. 2 I this paragraph, +1 ad -1 are assumed to occur i each cell with equal probability, so that E(w) is kl( 1 2 ). 3 If kl 100, x must be withi 5% of E(w). 100 was used i the hypothesis of 1.8 for the sake of cocreteess. Ay arbitrary costat c could have replaced 100, so that kl c. So log as c > 1, x is more limited tha w i the values it ca take. 4 kl >

4 4 QIAN ZHANG Remark 2.2. Vertices ca be visualized as poits ad a edge ca be visualized as a lie segmet coectig two poits. Defiitio 2.3. Cosider a graph G = (V, E) ad let deote G s maximum umber of possible edges, i.e. the umber of edges there ( would ) be if every vertex V were coected with every other vertex, so that =. E is the umber 2 of edges i the graph. The desity p of G is E. Defiitio 2.4. A bipartite graph Γ(L, R, E) is a graph cosistig of two sets of vertices L ad R such that a edge ca oly exist betwee a vertex i L ad a vertex i R. Call L the left set ad R the right set. Notatio 2.5. Give two sets of vertices V 1 ad V 2, E(V 1, V 2 ) deotes the set of edges betwee vertices i V 1 ad vertices i V 2. E(V 1, V 2 ) deotes the umber of elemets i E(V 1, V 2 ). Defiitio 2.6. Cosider a bipartite graph Γ(L, R, E) such that L = k ad R = l. Label the vertices i L with distict cosecutive atural umbers icludig 1 ad do the same for vertices i R. A adjacecy matrix (a ij ) of Γ is a k l matrix such that { 1 if i j, where i L ad j R a ij = 0 otherwise Remark 2.7. Let A be a k x l adjacecy matrix. (A T A) T = A T (A T ) T = A T A. Sice A T A is symmetric, it has l real eigevalues, deoted λ 1,..., λ l i decreasig order. A T A is positive semidefiite because x R l, x T A T Ax = Ax 2 0. Sice A T A is positive semidefiite, its eigevalues are oegative. Defiitio 2.8. A biregular bipartite graph Γ(L, R, E) is a bipartite graph where every vertex i L has the same degree s r ad every vertex i R has the same degree s c. Remark 2.9. E = L s r = R s c. Fact (Rayleigh Priciple) Let symmetric matrix A have eigevalues λ 1,..., λ i decreasig order. Defie the Rayleigh quotiet R A (x) = xt A x x T x. The λ 1 = R A (x). max x R, x 0 Lemma Let Γ(L, R, E) be a biregular bipartite graph with L = k ad R = l. Let each vertex i L have degree s r ad let each vertex i R have degree s c. Let A be the k l adjacecy matrix of Γ, ad let λ 1 be the largest eigevalue of A T A. The λ 1 = s r s c. Proof. Let r 1,..., r k be the row vectors of A. Recall that A has oly 1 or 0 for etries ad that each r i cotais s r 1 s, so dottig r i with some vector adds together s r compoets of that vector. A 1 l = 1 l 1 ( r 1 1 l 1,..., r k 1 l 1 ) l 2 where the last equality follows fro.9. = (s r,..., s r ) 1 k 2 l = ks2 r l = ( ks r )s r = s c s r l

5 QUASIRANDOMNESS AND GOWERS THEOREM 5 We have that A x 2 = s x 2 c s r whe x = 1 l l. If we could show that x R l, A x 2 s x 2 c s r, the we would have that A x 2 reaches its upper boud s x 2 c s r, so its max must be s c s r, ad by 2.10 ad 1.5, λ 1 = x T A T A x A x 2 max x R l, x 0 x T = max x x R l, x 0 x 2 = s c s r It remais to show that x R l, A x 2 x 2 s c s r. Let x 1,...x l deote the compoets of x. A x = ( r 1 x,..., r k x) T, so (2.12) A x 2 = k ( r i x) 2 r i x is the sum of s r compoets of x. Let x i1,..., x isr be the s r compoets of x that r i selects to sum. The r i x = sr x ij. r i x = s r x ij = (x i1,..., x isr ) 1 sr 1 1 sr 1 (x i1,..., x isr ) = s r s r (x ij ) 2 where the iequality follows from the Cauchy-Schwarz Iequality, so we have that ( r i x) 2 s r s r (x ij ) 2. Substitutig ito 2.12, (2.13) A x 2 s r k s r (x ij ) 2 Observe that the first summatio cycles through all the row vectors ad, for each row vector r i, the secod summatio cycles through the compoets of x chose by r i. Recall that A has s c 1 s i every colum, so i multiplyig A ad x, every compoet of x is selected by exactly s c row vectors. Hece, k s r l (x ij ) 2 = s c (x i ) 2 = s c x 2 Substitutig ito 2.13, A x 2 s r s c x 2, so x R l, A x 2 x 2 s c s r. Corollary Uder the assumptios of 2.11, 1 l l is a eigevector of A T A correspodig to eigevalue λ 1. Proof. Each etry of A 1 l 1 is the sum of a row of A, which is s r, so A 1 l 1 = s r 1 k 1. Similarly, A T 1 k 1 = s c 1 l 1. Hece, A T A 1 l 1 = A T (s r 1 k 1 ) = s r (A T 1 k 1 ) = s r s c 1 l 1 = λ 1 1 l, where the last equality follows by We have that A T A 1 l 1 = λ 1 1 l 1, so 1 l 1 is a eigevector of A T A correspodig to eigevalue λ 1. Notatio J deotes a matrix with 1 for every etry. Theore.16. (Quasiradomess Theorem) Suppose Γ(L, R, E) is a biregular bipartite graph with L = k ad R = l. Let the degree of every vertex i L be s r ad the degree of every vertex i R be s c. Let X L ad Z R, let p be the

6 6 QIAN ZHANG desity of Γ, let A be the k l adjacecy matrix of Γ, ad let λ i be the i th eigevalue of A T A i decreasig order. The E(X, Z) p X Z λ 2 X Z Proof. Let x be the icidece vector of X ad let z be the icidece vector of Z. 5 E(X, Z) = x T A z. Cosider the subgraph Γ(X, Z, E(X, Z)). If all vertices i X were coected with all vertices i Z, the umber of edges i the subgraph would be X Z = x T J k l z. E(X, Z) p X Z = x T A z p( x T J k l z) = x T (A pj k l ) z x T (A pjk l ) z = X (A pj k l ) z where the iequality follows by the Cauchy-Schwarz iequality. It remais to show that (A pj k l ) z λ 2 Z i.e. (A pj k l ) z 2 λ 2 Z = λ 2 z 2. (A pj k l ) z 2 = z T (A pj k l ) T (A pj k l ) z = z T (A T pj T k l)(a pj k l ) z = z T (A T A pa T J k l pj T k la + p 2 J T k lj k l ) z We will simplify A T A pa T J k l pjk l T A + p2 Jk l T J k l term-by-term. (Simplifyig Jk l T A) Γ is biregular: Every vertex i R is coected to s c vertices i L, so s c = E l, ad every vertex i L is coected to s r vertices i R, so s r = E k. Put aother way, the etries of each colum of A sum to s c ad the etries of each row of A sum to s r. p = E kl, so: s c = E l s r = E E kl = (kl) = pkl = pk l l E = kl (kl) = pkl k k = pl k Notice that each etry of Jk l T A is s c, which is pk, so Jk l T A = pkj l l. (Simplifyig A T J k l ) A T J k l = (Jk l T A)T = (pkj l l ) T = pkj l l, where the last equality holds because J l l is symmetric. (Simplifyig Jk l T J k l) Each etry of Jk l T J k l is the sum of a colum of J k l, which is k, so Jk l T J k l = kj l l. Substitutig i for Jk l T A, AT J k l, ad Jk l T J k l: A T A pa T J k l pj T k la + p 2 J T k lj k l = A T A p(pkj l l ) p(pkj l l ) + p 2 (kj l l ) = A T A p 2 kj l l M By 2.14, 1 is a eigevector of A T A to eigevalue λ 1 = s r s c = (pk)(pl) = p 2 kl. Sice J l l 1 = l 1, (p 2 kj l l ) 1 = p 2 k(j l l 1) = p 2 k(l 1) = (p 2 kl) 1 = λ 1 1. M 1 = A T A 1 p 2 kj l l 1 = λ 1 1 λ 1 1 = 0 = Cosider the submatrix of A that is the adjacecy matrix of Γ(X, Z, E(X, Z)). Call this submatrix B. I this setece, X refers to the set of rows of B ad Z refers to the set of colums of B, so 1.6 applies.

7 QUASIRANDOMNESS AND GOWERS THEOREM 7 so 1 is a eigevector of M correspodig to eigevalue 0. M = A T A p 2 kj l l = (A T A) T (p 2 kj l l ) T = (A T A p 2 kj l l ) T = M T. Sice M is a symmetric matrix, by the Spectral Theorem 3.22, there exists a orthogoal eigebasis (3.21) to M. Let e i be a vector i this orthogoal eigebasis, so M e i = u i e i, where u i R is a eigevalue of M ad u i s (aside from u 1 ) are idexed i decreasig order. Let e 1 1 l, so u 1 = 0. Sice the e i are orthogoal, 1 is orthogoal to e i, i 2. Notice that for i 2, each etry of J l l e i is 1 e i = 0, so J l l e i = 0. Hece, for i 2, M e i = (A T A p 2 kj l l ) e i = A T A e i p 2 k(j l l e i ) = A T A e i. For i 2, u i e i = M e i = A T A e i = λ i e i so u i = λ i for i 2. This implies that the largest eigevalue of M is λ 2, NOT λ 1 : Sice λ i s are ordered by size ad o u i = λ 1 for i 2 ad u 1 = 0, which is ot geerally equal to λ 1 = s r s c 0, o u i ever is λ 1. The ext largest value that a u i ca be is λ 2. (I particular, the largest eigevalue of M is u 2 = λ 2.). zt M z z T z max z z T M z z T z z By 2.10, the largest eigevalue of M is max T M z z z T z z T M z λ 2 z T z, ad z T z = z z = z 2, so z T M z λ 2 z 2. Recall, (A pj k l ) z 2 = z T (A pj k l ) T (A pj k l ) z = z T (A T A pa T J k l pj T k la + p 2 J T k lj k l ) z = z T M z λ 2 z 2 which is what we eeded to fiish the proof. = λ 2 The smaller λ 2 is, the closer E(X, Z) is to p X Z, so the closer E(X,Z) is to X Z p X Z X Z = p. Notice that E(X,Z) X Z is the desity of the bipartite subgraph formed by X ad Z, Γ(X L, Z R, E(X, Z)). Hece, the Quasiradomess Theorem says that the desity of Γ(X, Z, E(X, Z)) is approximately the desity of the larger graph Γ(L, R, E). Corollary Uder the same hypotheses as Theore.16, if p 2 X Z > λ 2, the E(X, Z) > 0. Proof. By 2.16, p 2 X Z > λ 2 p 2 ( X Z ) 2 > λ 2 X Z p X Z > λ 2 X Z p X Z λ 2 X Z > 0 E(X, Z) p X Z λ 2 X Z λ 2 X Z E(X, Z) p X Z p X Z λ 2 X Z E(X, Z) Combiig the above results, 0 < p X Z λ 2 X Z E(X, Z) 0 < E(X, Z) 2.1. How the Quasiradomess Theorem is a quasiradomess result. Defiitio A radom graph is a graph whose every pair of vertices is radomly assiged a edge. Pairs assigmets are idepedet of each other.

8 8 QIAN ZHANG Remark A radom bipartite graph is a radom graph such that ay two vertices i the same set have probability 0 of formig a edge. Cosider a radom situatio. Let G(L, R, E ) be a radom bipartite graph with L = k ad R = l, ad let each pair {l, r}, l L ad r R, have probability p of beig a edge. Let X L ad let Z R. Cosider the subgraph g(x, Z, E(X, Z )). The umber of pairs of vertices of g that ca form edges is X Z. Cosiderig the desigatio of edge a success, E(X, Z ), the umber of successes i X Z ( idepedet ) trials, would follow a biomial distributio: P ( E(X, Z ) = s) = X Z s p s (1 p) X Z s. E(X, Z ) would have expected value p X Z, so the desity of g, E(X,Z ) p X Z X Z = p. By the same argumet, P ( E = s) = X Z, would have expected value ( ) L R s p s (1 p) L R s, the expected value of E would be p L R, so the desity of G, E(L,R ) L R, would have expected value p. The desity of G ad the desity of g have the same expected value, but there is o guaratee that the desities be withi some rage of each other. The probability that the desities are wildly differet, say a desity of 0 ad a desity of 1, is ozero. Now cosider biregular bipartite graph Γ(L, R, E) described i the hypotheses of The Quasiradomess Theorem says that the desity of subgraph Γ(X L, Z R, E(X, Z)) must be withi some rage 6 of the desity of Γ(L, R, E), so i this sese oe ca expect the desity of Γ(X, Z, E(X, Z)) to be approximately the desity of Γ(L, R, E). Similarly, oe ca expect the desity of G to be approximately the desity of g (i the sese that their expected values are the same). However, ulike the desity of Γ(L, R, E) ad the desity of Γ(X, Z, E(X, Z)), the desity of G ad the desity of g are ot ecessarily withi some rage (other tha 1) of each other. Γ(L, R, E) is a quasiradom graph because it is like, but ot, a radom graph G(L, R, E ). Oe ca expect sufficietly large 7 subgraphs of Γ(L, R, E) to have desities similar to, yet differet from, desities of subgraphs of a radom graph. 3. Gowers theorem Defiitio 3.1. If F is a field ad V is a vector space over F, the GL(V) is the group of osigular liear trasformatios from V to V uder compositio. Defiitio 3.2. If d is a positive iteger, the the geeral liear group GL d (F) is the group of ivertible d d matrices with etries from F uder matrix multiplicatio. Defiitio 3.3. For a group G ad a iteger d 1, a d-dimesioal represetatio of G over F is a homomorphic map ϕ : G GL(V ) = GL d (F ), where V is a d-dimesioal vector space (V = F d ) ad F is a field. The degree of ϕ is d. 6 The rage is cotrolled by λ2 ad the sizes of X ad Z, ad could be less tha 1. The larger X ad Z are ad the smaller λ 2 is, the closer the desity of the subgraph is to the desity of the larger graph. 7 λ 2 X Z < 1

9 QUASIRANDOMNESS AND GOWERS THEOREM 9 Remark 3.4. To clarify, ϕ : G GL(V ) = GL d (R) is a represetatio of G over R. ϕ maps elemets of G to ivertible mappigs from R d to R d. Such mappigs correspod to d d ivertible matrices with etries from R. Defiitio 3.5. ϕ : G GL(V ) is a trivial represetatio if it maps every elemet of G to the idetity trasformatio. Theorem 3.6. (Gowers Theorem - GT) Let G be a group of order G ad let m be the miimum degree of otrivial represetatios of G over the reals. If X, Y, Z G ad X Y Z G 3 m, the x X, y Y, z Z s.t. xy = z. Corollary would still be true if its coclusio were replaced by XY Z = G Proof. Take X, Y, Z G such that X Y Z G 3 m. XY Z = G meas x X, y Y, z Z, g G s.t. xyz = g ad g G, x X, y Y, z Z, s.t. xyz = g. The first statemet holds by closure of G, so it remais to show the secod statemet. Take g G. Let Z = gz 1. By closure of G, Z G. Sice Z = Z, X Y Z G 3 m. By 3.6, x X, y Y, z Z s.t. xy = z xy(z 1 ) = z (z 1 ) = 1 xy(z 1 g) = g xyz = g Traslatig Gowers Theorem: Provig m Proves Gowers Theorem. Variables i this subsectio refer to those defied i the cotext of Γ(G 1, G 2, E): To prove 3.6, we take a graph theoretic view of it. Let G be a group. Let Γ(G 1, G 2, E) be a bipartite graph with two sets of vertices G 1 ad G 2, which are copies of G. Let there be a edge betwee g 1 G 1 ad g 2 G 2 oly if y Y G s.t. g 1 y = g 2, let A be the G G adjacecy matrix of Γ, let λ 2 be the secod largest eigevalue of A T A, let p be the desity of Γ, let X G 1, ad let Z G says that, for sufficietly large X ad Z, there is at least oe edge betwee a member of X ad a member of Z, i.e. E(X, Z) > 0. Curiously, which particular vertices are chose to costitute X ad Z is irrelevat to guarateeig a edge betwee them. Rather, the sizes of X ad Z are all that matter. I this graph theoretic view of Gowers Theorem, the hypotheses of the Quasiradomess Thrm hold. If p 2 X Z > λ 2 were to also hold, the by 2.17, 8 E(X, Z) > 0, provig Gowers Theorem. To traslate provig GT ito provig some other statemet, we use the followig results: Notatio 3.8. g 1 deotes ay elemet of G 1 ad g 2 deotes ay elemet of G 2. Lemma 3.9. The degree of every vertex of Γ(G 1, G 2, E) is Y Proof. We will show that every vertex i G 1 has degree Y ad every vertex i G 2 has degree Y, so every vertex of Γ has degree Y. Claim: Every g 1 has degree Y. Sice G is a group, by closure, g 1 G 1 = G ad y Y G, g 1 y G = G 2 so g 1 y = g 2. For each g 1, multiplicatio with each y yields Y distict products i G 2. Sice g 1 ad g 2 form a edge iff y Y s.t. g 1 y = g 2, g 1 ca form o other edges, so the degree of every g 1 is Y shows that Γ is biregular

10 10 QIAN ZHANG Claim: Every g 2 has degree Y, i.e. every g 2 has Y preimages i G 1 : y Y, uique g 1 G 1 s.t. g 1 y = g 2. Take y Y G. Sice G is a group, y 1 G. Take g 2 G 2 = G. By closure, g 2 y 1 G = G 1 so g 1 = g 2 y 1. Corollary E = G Y Proof. Every g 1 G 1 forms Y edges, ad there are G g 1 s, so E = G Y Fact If A is a real matrix with eigevalues λ 1,..., λ, the T r(a) = λ i Notatio λ i deotes oe of the G eigevalues of A T A: { λ 1,..., λ G }, listed i decreasig order. m i deotes the multiplicity of λ i. Corollary λ 2 < T r(at A) Proof. By 3.11, T r(a T A) = G λ i = λ 1 + λ > λ 2, where the last iequality follows from A T A havig oegative eigevalues (by 2.7) ad from λ 1 > 0 9. Lemma T r(a T A) = E Proof. Let c 1,..., c G be the colum vectors of A ad let c ij deote a compoet of c j. G T r(a T A) = c j c j = ( c 2 ij) = ( c ij ) G The last equality follows because etries of A are either 0 or 1, so c 2 ij = c ij. The double summatio adds all the etries of A, hece couts the umber of edges of Γ(G 1, G 2, E). I more detail: The secod summatio gives the degree of a particular g 2. The first summatio cycles through all vertices i G 2. Hece, the double summatio couts all the edges that vertices i G 2 are members of, so it couts all the edges of Γ. Corollary λ 2 < G Y Proof. λ 2 < T r(at A). The first iequality holds by 3.13, the secod equality holds by 3.14, ad the third equality holds by Remark p = G Y = E = G Y G G = Y G G G G, where the first equality follows from 3.10 ad 2.3. Propositio To prove Gowers Theorem, it remais to show that m. Proof. From 3.15, we have that λ 2 < G Y. If we could show that G Y p 2 X Z, the λ 2 < p 2 X Z, fulfillig the hypotheses of 2.17 ad reachig the coclusio of Gowers Theorem 10. I other words, to prove GT, it remais to prove p 2 X Z. G Y 9 By 2.11, λ1 = s rs c. Recall that s r is the sum of each row of A ad s c is the sum of each colum of A. Here, we assume that Γ is a otrivial graph, i.e. it actually has edges, so s r > 0 ad s c > 0. Hece, λ 1 > its graph theoretic iterpretatio: E(X, Z) > 0

11 QUASIRANDOMNESS AND GOWERS THEOREM 11 G Y p 2 X Z G Y ( ) 2 Y G X Z G 3 X Y Z, where the first iff follows from To prove GT it remais to prove G 3 X Y Z. Give GT s hypothesis X Y Z G 3 m, if we could show m, the X Y Z G 3. Hece, to prove GT it remais to prove m Provig m. Recall that is the multiplicity of λ 2 ad m is the miimum degree of otrivial represetatios of G over R i.e. the smallest dimesio of a real vector space i which G has otrivial represetatio. To show that m, we will eed some prelimiary defiitios ad results. Defiitio Let V be a d-dimesioal vector space. U V is ivariat uder ϕ : G GL(V ) if for all g G, U is ivariat uder ϕ(g), i.e. u U, g G, ϕ(g)u U. Every mappig that ϕ associates with a elemet of G maps U to U. Defiitio If λ F ad A is a x matrix over F, the the eigespace to eigevalue λ is U λ = { x F s.t. A x = λ x}. A member of the eigespace is called a eigevector correspodig to λ. Lemma If AB = BA, the every eigespace of A is ivariat uder B. Proof. Let U λ be a eigespace of A. We wat to show that x U λ, B x U λ. Sice x U λ, A x = λ x, so AB x = BA x = B(λ x) = λb x. Defiitio A eigebasis of a matrix A is a set of eigevectors of A that forms a basis for the domai of the liear trasformatio correspodig to A. Theorem (Spectral Theorem) Every real symmetric matrix has a orthogoal eigebasis. Notatio Give mappig f : A B ad C A, f C deotes the mappig that is the same as f, except with domai restricted to C. Hom(A,B) deotes the set of homomorphisms from A to B. Propositio Let A = A T be a real d d matrix, ad let G be a group. Let m = mi{s : φ otrivial Hom(G, GL s (R))}, i.e. m is the miimum degree of otrivial represetatios of G over the reals. Let ϕ Hom(G, GL d (R)) be otrivial. Suppose that A commutes with all matrices i GL d (R). The there is a eigevalue of A with multiplicity at least m. Proof. By 3.22, we ca choose a particular eigebasis of A. Call this basis B A = { e 1,..., e d }. Sice ϕ is otrivial, we ca pick g 0 G, such that ϕ(g 0 ) is ot the idetity matrix. Let ψ : R d R d be the uique liear map whose trasformatio matrix with respect to B A is ϕ(g 0 ). Sice ϕ(g 0 ) is ot the idetity matrix, ψ is ot the idetity map o R d. Sice A commutes with every elemet of GL d (R), by 3.20, for every g G ad every eigespace U λ of A, U λ is ivariat uder ϕ(g). Hece, ϕ Uλ : g ϕ(g) Uλ is ϕ Uλ : G GL(U λ ), a dim(u λ )-dimesioal represetatio of G. I particular, A commutes with ϕ(g 0 ), so by 3.20, ψ seds each eigespace of A to itself. ψ caot act as the idetity o every U λ, because if it did, the v R d, v = d α i e i where α i R, ad

12 12 QIAN ZHANG ψ( v) = ψ( d α i e i ) = d α i ψ( e i ) = d α i e i = v so ψ would act as the idetity o R d, which is cotrary to the choice of ψ. We ve show by cotradictio that there must be a eigespace U λ of A such that ψ : U λ U λ is ot the idetity map. Because ψ U is ot the idetity λ map, ϕ(g 0 ) Uλ is ot the idetity matrix, so ϕ U : G GL(U λ ) is a otrivial λ represetatio of G. By defiitio, m is the miimum degree of otrivial represetatios of G, so the degree of ϕ U (which is the dimesio of U λ λ ) is at least m. Sice A is symmetric, the dimesio of U λ is the multipliticy of λ, so the multiplicity of λ is at least m. Defiitio σ : V V is a permutatio o set V if it is a bijectio. Defiitio Cosider a graph G = (V, E). A graph automorphism is a mappig σ : V V that preserves adjacecy, i.e. i, j V, i j σ(i) σ(j) Remark A graph automorphism for a bipartite graph Γ(V 1, V 2, E) cosists of permutatios σ 1 : V 1 V 1 ad σ 2 : V 2 V 2 s.t. v 1 V 1 ad v 2 V 2, v 1 v 2 σ 1 (v 1 ) σ 2 (v 2 ). Defiitio P (σ) is a permutatio matrix of permutatio σ if { 1 if σ(i) = j P (σ) ij = 0 otherwise. Lemma Let Γ(V 1, V 2, E) be a biregular bipartite graph, let A be its adjacecy matrix, let σ 1 be a permutatio of V 1, ad let σ 2 be a permutatio of V 2. The σ 1 ad σ 2 costitute a bipartite graph automorphism iff P (σ 1 )A = AP (σ 2 ) Proof. The claim is that i V 1, j V 2, i j σ 1 (i) σ 2 (j) P (σ 1 )A = AP (σ 2 ) We will traslate the right-had side ito some other statemet. By defiitio, P (σ 1 )A = AP (σ 2 ) i, j, [P (σ 1 )A] ij = [AP (σ 2 )] ij. For all i, j, [AP (σ 2 )] ij = L l=1 A ilp (σ 2 ) lj. Notice that cells of A ad cells of P oly take values 1 or 0, so terms of the sum are either 1 or 0. The summatio is equivalet to summig oly the terms that are 1. For a term to be 1, A il ad P (σ 2 ) lj must both be 1. By defiitio, A il = 1 iff i l, ad P (σ 2 ) lj = 1 iff σ 2 (l) = j. Hece, A il P (σ 2 ) lj = 1 iff i l ad σ 2 (l) = j, so L A il P (σ 2 ) lj = A ilp (σ 2 ) lj. l=1 l s.t. i l=σ 1 2 (j) Multiple l s ca be adjacet to i, but sice σ 2 is oe-to-oe, oly oe l ca equal σ 1 2 (j), so [AP (σ 2 )] ij = l s.t. i l=σ 1 2 (j) A ilp (σ 2 ) lj = { 1 if i σ 1 2 (j) 0 otherwise

13 QUASIRANDOMNESS AND GOWERS THEOREM 13 For all i,j,[p (σ 1 )A] ij = K P (σ 1 ) ik A kj. The terms of this sum are either 1 or 0, k=1 so the sum is equivalet to summig oly the terms that are 1. For a term to be 1, P (σ 1 ) ik ad A kj must both be 1. By defiitio, P (σ 1 ) ik = 1 iff σ 1 (i) = k, ad A kj = 1 iff k j. Hece, P (σ 1 ) ik A kj = 1 iff σ 1 (i) = k ad k j, so K P (σ 1 ) ik A kj = P (σ 1 ) ik A kj k=1 k s.t. σ 1(i)=k j. Multiple k could be adjacet to j, but sice σ 1 is oe-to-oe, oly oe k = σ 1 (i). Hece, the summatio ca have oly oe term that is 1, so { 1 if σ1 (i) j [P (σ 1 )A] ij = P (σ 1 ) ik A kj = 0 otherwise k s.t.σ 1(i)=k j For all i,j [P (σ 1 )A] ij = [AP (σ 2 )] ij iff the cells are both 1 or both 0 iff ( σ 1 (i) j ad i σ2 1 (j)) or ( (σ 1 (i) j) ad (i σ2 1 (j))). Hece, σ 1 (i) j is equivalet to i σ2 1 (j). To summarize, P (σ 1 )A = AP (σ 2 ) meas i, j, σ 1 (i) j iff i σ2 1 (j), so the lemma says: i V 1, j V 2, i j σ 1 (i) σ 2 (j) i V 1, j V 2, σ 1 (i) j i σ 1 2 (j) ( )Suppose (3.30) i V 1, j V 2, i j σ 1 (i) σ 2 (j). We wat to show σ 1 (i) j i σ 1 2 (j). (3.31) σ 1 (i) j σ 1 (i) σ 2 (σ2 1 (j)) i σ 1 2 (j) where the last equivalece comes from the directio of 3.30 ( )Suppose (3.32) i V 1, j V 2, σ 1 (i) j i σ 1 2 (j) We wat to show i j σ 1 (i) σ 2 (j). (3.33) i j i σ 1 2 (σ 2(j)) σ 1 (i) σ 2 (j) where the last equivalece comes from the of 3.32 Claim Let σ be a permutatio ad let P be its permutatio matrix. P T = P 1. Proof. The claim is that P P T = P T P = I. Recall that P (σ) ij is 1 if σ(i) = j ad is 0 otherwise. Sice σ is a fuctio, every row vector of P has oly oe etry that is 1. Sice σ is bijective, every colum vector of P has oly oe etry that is 1. No two row vectors ca have same the

14 14 QIAN ZHANG same compoet be 1, because if there were two such row vectors, there would be a colum vector with more tha oe 1-etry, cotradictig that every colum vector has oly oe 1-etry. Similarly, o two colum vectors ca have the same compoet be 1. Hece, every pair of distict row vectors of P is orthogoal ad every pair of distict colum vectors of P is orthogoal. Let the rows of P be r 1,..., r. (P P T ) ii = r i r i = r ij = 1, sice every row vector has oly oe etry that is 1. For i j, (P P T ) ij = r i r j = 0 sice row vectors are orthogoal. Hece, P P T = I. Let the colums of P be c 1,..., c. (P T P ) ii = c i c i = c ij = 1, sice every colum vector has oly oe etry that is 1. For i j, (P P T ) ij = c i c j = 0, sice colum vectors are orthogoal. Hece, P T P = I. Corollary Uder the assumptios of 3.29, P (σ 2 )A T = A T P (σ 1 ) by 3.29 ad 3.34 Lemma Let Γ(V 1, V 2, E) be a bipartite graph with adjacecy matrix A. Let σ 1 be a permutatio o V 1 ad let σ 2 be a permutatio o V 2. Let σ 1 ad σ 2 costitute a graph automorphism. The P (σ 2 ) commutes with A T A. Proof. P (σ 2 ) 1 A T AP (σ 2 ) = P (σ 2 ) T A T (I k k )AP (σ 2 ) = P (σ 2 ) T A T (P (σ 1 )P (σ 1 ) 1 )AP (σ 2 ) = P (σ 2 ) T A T (P (σ 1 )P (σ 1 ) T )AP (σ 2 ) = (P (σ 2 ) T A T P (σ 1 ))(P (σ 1 ) T AP (σ 2 )) = (P (σ 2 ) 1 A T P (σ 1 ))(P (σ 1 ) 1 AP (σ 2 )) = A T A The last equivalece follows from 3.29 ad We have P (σ 2 ) 1 A T AP (σ 2 ) = A T A, so A T AP (σ 2 ) = P (σ 2 )A T A. Now cosider the particular bipartite graph Γ(G 1, G 2, E) ivolved i the proof of Gowers Theorem. A is its adjacecy matrix, which is a G x G real matrix, so A T A is a G x G real matrix. Let λ 2 be the secod largest eigevalue of A T A. Let σ 1 be a permutatio of G 1, let σ 2 be a permutatio of G 2 ad let σ 1 ad σ 2 costitute a graph automorphism. Let ϕ : g P (σ 2 ) be a otrivial represetatio of G, i.e. let ϕ map some g to a P (σ 2 ) that is ot the idetity matrix. Let ψ be the liear trasformatio correspodig to this P (σ 2 ). Usig a argumet similar to that i the proof of Propositio 3.27, we will show that m. Remark Recall U λ2 (A T A) { x R G s.t. A T A x = λ 2 x } Propositio m Proof. P (σ 2 ) is ot the idetity matrix, so ψ does ot act as the idetity o R G. Suppose we could show that ψ does ot act as the idetity o U λ2 (A T A). The P (σ 2 ) Uλ2 (AT A) would ot be the idetity matrix. Hece, ϕ Uλ2 (AT A) : G P (σ 2 ) Uλ2 (AT A) would be a otrivial represetatio of G, so its degree would be at least the miimum degree of a otrivial represetatio of G i.e. m. By 3.36, each P (σ 2 ) commutes with A T A, so by 3.20, U λ2 (A T A) is ivariat uder every P (σ 2 ), so P (σ 2 ) Uλ2 (AT A) = GL(U λ2 (A T A)), so ϕ Uλ2 (AT A) : G

15 QUASIRANDOMNESS AND GOWERS THEOREM 15 GL(U λ2 (A T A)). The dimesio of U λ2 (A T A) is the degree of ϕ Uλ2 (A T A), which we already showed is at least m. Sice A T A is symmetric, = dim(u λ2 (AT A)), which is at least m, so m. It remais to show that ψ does ot act as the idetity o U λ2 (A T A). The oly way a ψ that is ot the idetity trasformatio ca act as the idetity o U λ2 (A T A) is if each vector i U λ2 (A T A) has idetical compoets, i.e. is a multiple of 1. By 2.14, A T A 1 = λ 1 1 λ 2 1, so for c R, A T A(c 1) λ 2 (c 1) so o multiple of 1 is i U λ2 (A T A), so ψ caot act as the idetity o U λ2 (A T A) fiishes the proof of Gowers Theorem. Ackowledgemets Thaks to my metors Irie Peg ad Marius Beceau for their help. Refereces [1] L. Babai. Class Notes: Discrete Math - First 4 Weeks. laci/reu07/. [2] Dummit ad Foote. Abstract Algebra 3rd editio. Joh Wiley ad Sos Ic [3] J. A. Rice. Mathematical Statistics ad Data Aalysis. Duxbury Press