The sum-product theorem and applications Avi Wigderson School of Mathematics Institute for Advanced Study

Plan of the talk Background and statement of the S-P Theorem Applications of the Theorem to: -- Combinatorial Geometry -- Number Theory -- Group Theory -- Extractor Theory Sketch of the proof of S-P Theorem -- Balog-Szemeredi-Gowers Lemma -- Plunneke-Rusza Inequalities

Sum-Product in the Reals F a field, A  F A+A = { a+b : a,b  A } A  A = { a  b : a,b  A } A={1,2,3,…k} then |A+A| < 2|A| A={1,2,4,…2 k } then |AxA| < 2|A| Is there a set A for which both |A+A|, |A  A| small? Thm[ES] F=R.  >0  A either |A+A|>|A| 1+  or |A  A|>|A| 1+ 

Sum-Product in finite fields Thm[ES] F=R.  >0  A either |A+A|>|A| 1+  or |A  A|>|A| 1+  Can this be true in a finite field? But if A=F or too big… Assume |A|<|F|.9 But if A is a subfield… Assume F has no subfields Thm:[BKT,K] F=F p for prime p.  >0  A, |A|<|F|.9 either |A+A|>|A| 1+  or |A  A|>|A| 1+  Variants for larger A, small subfields, rings, etc

Applications / Implications -- Combinatorial Geometry -- Number Theory -- Group Theory -- Extractor Theory

Combinatorial Geometry P – a set of points in F 2 |P|=n L – a set of lines in F 2 |L|=n I = { (p,l) : point p is on line l } = incidences BEFORE S-P THM Trivial: Any plane |I| < n 3/2 Thm[ST,E] F=Reals |I| < n 3/11 USING S-P THM Thm[BKT] F=F p |I| < n 3/2- 

Number Theory F=F p G multiplicative subgroup of F * S(a,G) =  g  G  ag Fourier coefficient at a S(G) = max { |S(a,G)| : a  F * } Trivial S(G)  |G|. Want S(G)  |G| 1- . BEFORE S-P THM |G| > p 1/2 [W]… p 3/7 [HB]… p 1/4 [KS] USING S-P THM Thm[BK] |G| > p  implies S(G)  |G| 1-  (  ). Thm[BGK] |A| > p  implies S(A k )  |G| 1-  (  ) k=k(  )

Group Theory H a finite group, T a (symmetric) set of generators of H. Cay(H;T) the Cayley graph: g  h iff gh -1  T. Diam(H;T) the diameter of Cay(H;T) (H;T): 2 nd e-val of random walk on Cay(H;T) Cay(H;T) expander  (H;T) < 1-   diam(H;T) < O(log |H|) H=SL(2,p), the group of 2  2 invertible matrices over F p BEFORE S-P THM Thm[S,LPS,M] Few T’s for which Cay(H;T) expands USING S-P THM Thm[H]  T, diam(H;T) < polylog(|H|) Thm[BG] |T|=2, random, then Cay(H;T) expands whp not cyclic in SL(2,Z), then Cay(H;T) expands

Extractors & Dispersers S a class of probability distributions on {0,1} n X  S is often called a “weak source” of randomness f :{0,1} n  {0,1} m which for all X  S satisfies -- |f(X)| > (1-  )2 m is called an (S,  )-disperser -- |f(X) – U m | 1 <  is called an (S,  )-extractor Existence of f is a Ramsey/Discrepancy Theorem Want Explicit (polytime computable) f. Important research area with many applications.

Affine sources f :{0,1} n  {0,1} m which for all X  S satisfies -- |f(X)| > (1-  )2 m is called an (S,  )-disperser -- |f(X) – U m | 1 <  is called an (S,  )-extractor S=L k : Affine subspaces of (F 2 ) n of dimension  k. f optimal if m =  (k) and  = 2  (-k) BEFORE S-P THM Exists: Optimal affine extractor  k>2log n Explicit: Optimal affine extractor  k> n/2 USING S-P THM [BKSSW] Explicit affine disperser with m=1  k>  n [B] Explicit optimal affine extractor  k>  n [GR] Extractors for large fields with low dimension

Two independent sources f :{0,1} n  {0,1} m which for all X  S satisfies -- |f(X)| > (1-  )2 m is called an (S,  )-disperser (m=1 : bipartite Ramsey Graph) -- |f(X) – U m | 1 <  is called an (S,  )-extractor S=I k : {  (X 1,X 2 ): H  (X i )  k}. X i  {0,1} n/2 independent f optimal if m =  (k) and  = 2  (-k) BEFORE S-P THM [E] Exist optimal 2-source extractor  k>2log n [CG,V] Explicit optimal 2-source extractor  k> n/2 USING S-P THM [P/B] Explicit optimal 2-sourse dis /ext k>.4999n [BKSSW] Explicit 2-s disperser with m=1  k>  n [BRSW] Explicit 2-s disperser with m=1  k> n 

Statistical version of S-P Thm A distribution on F p, H 0 (A) = log |supp(A)| H 2  H Shannon  H 0 H 2 (A) = -log ||A|| 2  (  H  (A) ) [BKT, K] H 0 (A) (1+  )H 0 (A) or H 0 (A  A) > (1+  )H 0 (A) Want H 2 (A) (1+  )H 2 (A) or H 2 (A  A) > (1+  )H 2 (A) False: A = (Arithmetic prog + Geometric prog)/2 [BKT, K] H 0 (A) (1+  )H 0 (A) [BIW] H 2 (A) (1+  )H 2 (A) up to exponential L1 error.

Statistical S-P & Extractors A,B,C indep dist. on F p, r (A) = H 2 (A) / (log p) r = min { r(A), r(B), r(C) } [BIW] r (1+  )r r >.9  r(A  B+C) = 1 [Z] Conjectured extractor from many indep sources f 1 (A 1,A 2,A 3 ) = A 1  A 2 +A 3 f t+1 (3 t+1 sources) = f 1 ( f t (3 t ), f t (3 t ), f t (3 t ) ) S = {(A 1,A 2,…A c ) indep sources on {0,1} n/c, H 2 (A i ) > k} [BIW] Opt explicit extractor for k=  n, c=poly(1/  ) [R] Opt explicit extractor for k=n , c=poly(1/  )

Statistical S-P & Condensers A,B,C indep dist. on F p, r(A) = H 2 (A) / (log p) r = min { r(A), r(B), r(C) } [BIW] r (1+  )r X distribution on {0,1} n r(X) = H 2 (X) / n f :{0,1} n  ({0,1} m ) c is a condenser if  X, r(X) (1+  )r(X) [BKSSW] X=(A,B,C)  A, B, C, A  B+C condenser Iterating… r=  .9 with c=poly(1/  ) m=  (n)

Proof of the S-P theorem Thm [BKT] F=F p.  0  A, |A|=|F|  either |A+A|>|A| 1+  or |A  A|>|A| 1+  Proof[BIW]  =  (  ) Rational expression R(A) e.g (A+A-A  A)/(A  A  A) = {(a 1 +a 2 -a 3 a 4 )/(a 5 a 6 a 7 )} Lemma 1:  R 0  A |R 0 (A)|>|A| 1+  Lemma 2: |A+A|>|A| 1+  and |A  A|<|A| 1+  then  R  c=c(R) |R(A)|<|A| 1+c   B |B|>|A| 1-c  and |R(B)|<|B| 1+c  Lemma 1 + Lemma 2 imply Thm

Proof of the Lemma 1 Lemma 1:  R 0  A, |A|=|F|  we have |R 0 (A)|>|A| 1+  Proof: Pigeonhole principle &  k  N, |F| 1/k  N A’ = (A-A)/(A-A) |A’|=|F|  ’ R 0 (A) = A’’ = (A’-A’)/(A’-A’) |A’’|=|F|  ’’ Claim:  (1/(k+1),1/k)  |A| k < |F| < |A| k+1   ’ > 1/k   ’’ > 1/(k-1) >  (1+  ) Proof: Assume  ’<1/k. Set 1=s 0,s 1,…,s k  F s.t.  j s j  s 0 A’ + s 1 A’ +… + s j-1 A’ Define g:A k+1  F by g(x 0,x 1,…,x k ) =  s i x i |A k+1 |>|F|.  x  y  s i x i =  s i y i j largest s.t. x j  y j s j =  i<j s i (x i- y i )/(x j- y j )  s 0 A’+ s 1 A’+… +s j-1 A’ #

Ingredients for Lemma 2 G Abelian group. A  G,  >0 arbitrary. Thm[R] |A+A| < |A| 1+   |A-A| < |A| 1+2  Cor: R(A) large then P(A) large for a polynomial P Thm[P,R] |A+A| < |A| 1+   |A+kA| < |A| 1+k  Thm[BS,G] ||A+A|| -1 < |A| 1+    A’  A, |A’|> |A| 1-5  but |A’+A’| < |A’| 1+5  All proofs: Graph Theory

Conclusions & Problems -Sum-Product Theorem is fundamental! -Has many variants and extensions (e.g. to rings) -Has many more applications -What is the best  in the S-P theorem? - in the Reals believed to be 1 - in finite fields cannot exceed 1/2 -2-source extractor for entropy <.4999n -2-source disperser for entropy << n o(1)