The sum-product theorem and applications

1
The sum-product theoremand applications

• Avi Wigderson
• School of Mathematics
• Institute for Advanced Study

2
Plan of the talk

• Background and statement of the S-P Theorem
• Applications of the Theorem to
• -- Combinatorial Geometry
• -- Analysis/PDE
• -- Number Theory
• -- Group Theory
• -- Extractor Theory
• Sketch of the proof of S-P Theorem
• -- Balog-Szemeredi-Gowers Lemma
• -- Plunneke-Rusza Inequalities

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Sum-Product in the Reals

• F a field, A?F
• AA ab a,b?A
• A?A a?b a,b?A
• A1,2,3,k then AA lt 2A
• A1,2,4,2k then AxA lt 2A
• Is there a set A for which both AA, A?A
  small?
• ThmES FR.
• ??gt0 ?A either AAgtA1? or A?AgtA1?

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Sum-Product in finite fields

• ThmES FR.
• ??gt0 ?A either AAgtA1? or A?AgtA1?
• Can this be true in a finite field?
• But if AF or too big Assume AltF.9
• But if A is a subfield Assume no subfields
• ThmBKT,K FFp p prime. ??gt0 ?A, AltF.9
• either AAgtA1? or A?AgtA1?
• Variants for small subfields, rings, etc.

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Applications / Implications

• -- Combinatorial Geometry
• -- Analysis/PDE
• -- Number Theory
• -- Group Theory
• -- Extractor Theory

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Combinatorial Geometry

• P a set of points in F2 Pn
• L a set of lines in F2 Ln
• I (p,l) point p is on line l
  incidences
• BEFORE S-P THM
• Trivial Any plane I lt n3/2
• ThmST,E FReals I lt n4/3
• USING S-P THM
• ThmBKT FFp I lt n3/2-?

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Analysis/PDE

• - Divergence of Fourier series in Lp spaces (p?2)
• Instability of solutions to wave equations
• Kakeya Find the least area of a set S in R2
  containing a unit segment in every direction?
• Besicovitch As small as you wish!
• Questions Other measures (Hausdorff)
• Higher dimensions (Rd)

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Besicovitch construction
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Kakeyas problemfinite fields

• S ? (Fp)d a Kakeya set, if S contains a line in
  every possible direction (for large p).
• B(d) is the smallest r such that ?S, S?(pr)
• Conjecture B(d) d
• BEFORE S-P THM
• Trivial B(d) ? d/2
• ThmW B(d) ? d/2 1
• USING S-P THM
• ThmBKT B(d) ? d/2 1 10-10

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Number Theory

• FFp 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) ? G1-? .
• BEFORE S-P THM
• G gt p1/2 W p3/7 HB p1/4 KS
• USING S-P THM
• ThmBK G gt p? implies S(G) ? G1-?(?) .
• ThmBGK A gt p? implies S(Ak(?) ) ? Ak1-?(?)

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Group Theory

• H a finite group, T a (symmetric) set of
  generators of H.
• Cay(HT) the Cayley graph g?h iff gh-1?T.
• Diam(HT) the diameter of Cay(HT)
• ?(HT) 2nd e-val of random walk on Cay(HT)
• Cay(HT) expander ? ?(HT) lt 1- ? ?
  diam(HT) lt O(log H)
• HSL(2,p), the group of 2?2 invertible matrices
  over Fp
• BEFORE S-P THM
• ThmS,LPS,M Few Ts for which Cay(HT) expands
• USING S-P THM
• ThmH ?T, diam(HT) lt polylog(H)
• ThmBG T2, random, then Cay(HT) expands
  whp
• ltTgt not cyclic in SL(2,Z), then Cay(HT)
  expands

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Extractors Dispersers

• S a class of probability distributions on 0,1n
• X?S is often called a weak source of
  randomness
• f 0,1n ? 0,1m which for all X?S satisfies
• -- f(X) gt (1-?)2m is called an
  (S,?)-disperser
• -- f(X) Um1 lt ? is called an
  (S,?)-extractor
• Existence of f is a Ramsey/Discrepancy Theorem
• Want Explicit (polytime computable) f.
• Important research area with many applications.

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Affine sources

• f 0,1n ? 0,1m which for all X?S satisfies
• -- f(X) gt (1-?)2m is called an
  (S,?)-disperser
• -- f(X) Um1 lt ? is called an
  (S,?)-extractor
• SLk Affine subspaces of (F2)n of dimension ??k.
• f optimal if m ?(k) and ? 2?(-k)
• BEFORE S-P THM
• Exists Optimal affine extractor
  ?kgt2log n
• Explicit Optimal affine extractor ?kgt
  n/2
• USING S-P THM
• BKSSW Explicit affine disperser with m1 ?kgt ?n
  
• B Explicit optimal affine extractor
  ?kgt ?n
• GR Extractors for large fields with low
  dimension

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Two independent sources

• f 0,12n ? 0,1m which for all X?S
  satisfies
• -- f(X) gt (1-?)2m is called an
  (S,?)-disperser (m1 bipartite Ramsey Graph)
• -- f(X) Um1 lt ? is called an
  (S,?)-extractor
• SIk ?(X1,X2) H?(Xi)?k. Xi?0,1n
  independent
• f optimal if m ?(k) and ? 2?(-k)
• BEFORE S-P THM
• E Exist optimal 2-source extractor
  ?kgt2log n
• CG,V Explicit optimal 2-source extractor ?kgt
  n/2
• USING S-P THM
• B Explicit optimal 2-sourse extractor kgt
  .4999n
• BKSSW Explicit 2-s disperser with m1 ?kgt ?n
• BRSW Explicit 2-s disperser with m1 ?kgt n?

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Statistical version of S-P Thm

• A distribution on Fp, H0(A) log supp(A)
• H2 ? HShannon ? H0 H2(A) -log A2 ? (
  ? H?(A) )
• BKT, K H0(A) lt .9log p ? H0(AA) gt
  (1?)H0(A)
• or
  H0(A?A) gt (1?)H0(A)
• Want H2(A) lt .9log p ? H2(AA) gt
  (1?)H2(A)
• or
  H2(A?A) gt (1?)H2(A)
• False A (Arithmetic prog Geometric prog)/2
• BKT, K H0(A) lt .9log p ? H0(A?AA) gt
  (1?)H0(A)
• BIW H2(A) lt .9log p ? H2(A?AA) gt
  (1?)H2(A)
• 
  up to exponential L1 error.

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Few Independent Sources

• A,B,C indep dist. on Fp, r (A) H2(A) / (log
  p)
• r min r(A), r(B), r(C)
• BIW r lt .9 ? r(A?BC) gt (1?)r
• r gt .9 ? r(A?BC) 1
• Extractor from several independent sources
• f1(A1,A2,A3) A1?A2A3
• ft1 (3t1 sources) f1 ( ft(3t),
  ft(3t), ft(3t) )
• S (A1,A2,Ac) indep sources on 0,1n, H2(Ai)
  gt k
• BIW Opt explicit extractor for k?n,
  cpoly(1/?)
• R Opt explicit extractor for kn?,
  cpoly(1/?)

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Statistical S-P Condensers

• A,B,C indep dist. on Fp, r(A) H2(A) / (log
  p)
• r min r(A), r(B), r(C)
• BIW r lt .9 ? r(A?BC) gt (1?)r
• X distribution on 0,1n r(X) H2(X) / n
• f 0,1n ? (0,1m)c is a condenser
• if ?X, r(X) lt .9 ? ??i r(f(X)i) gt
  (1?)r(X)
• BKSSW X(A,B,C) ? A, B, C, A?BC
  condenser
• Iterating r? ? .9 with cpoly(1/?) m?(n)

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Proof of the S-P theorem

• Thm BKT FFp. ??lt.9 ??gt0 ?A, AF?
• either AAgtA1? or A?AgtA1?
• ProofBIW ? ?(?)
• Rational expression R(A) e.g (AA-A?A)/(A?A?A)
• 
  (a1a2-a3a4)/(a5a6a7)
• Lemma 1 ?R0 ?A R0(A)gtA1?
• Lemma 2 AAltA1? and A?AltA1? then
• ?R ?cc(R) R(A)ltA1c?
• ?B BgtA1-c? and R(B)ltB1c?
• Lemma 1 Lemma 2 imply Thm

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Proof of the Lemma 1

• Lemma 1 ?R0 ?A, AF? we have
  R0(A)gtA1?
• Proof Pigeonhole principle ?k?N, F1/k?N
• A (A-A)/(A-A) AF?
• R0(A) A (A-A)/(A-A) AF?
• Claim ??(1/(k1),1/k) (?? Ak lt F lt Ak1
  )
• ? ? gt 1/k ? ? gt 1/(k-1) gt
  ?(1?)
• Proof Assume ?lt1/k. Set 1s0,s1,,sk?F s.t.
• ?j sj ? s0A s1A sj-1A
• Define gAk1 ? F by g(x0,x1,,xk) ?sixi
• Ak1gtF. ?x?y ?sixi ?siyi j largest
  s.t. xj ?yj
• sj ?iltj si(xi-yi)/(xj-yj) ? s0A s1A
  sj-1A

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Ingredients for Lemma 2

• G Abelian group. A?G, ?gt0 arbitrary.
• ThmR AA lt A1? ? A-A lt A12?
• Cor R(A) large then P(A) large for a polynomial
  P
• ThmP,R AA lt A1? ? AkA lt A1k?
• ThmBS,G AA-1 lt A1? ?
• ?A ? A, Agt A1-5? but AA lt
  A15?
• All proofs Graph Theory

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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 lt .4999n
• 2-source disperser for entropy ltlt no(1)