Jean Bourgain Institute for Advanced Study Princeton, NJ 08540

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1 Jean Bourgain Institute for Advanced Study Princeton, NJ

2 ADDITIVE COMBINATORICS SUM-PRODUCT PHENOMENA Applications to: Exponential sums Expanders and spectral gaps Invariant measures Pseudo-randomness 2

3 SUM-PRODUCT THEOREM IN F p = Z/pZ Theorem. (BKT, BGK). For all ε > 0, there is δ > 0 such that if A F p and A < p 1 ε, then A + A + A.A > c A 1+δ Proof based on Plunnecke Ruzsa theory of set addition Extensions to: Arbitrary finite fields F p r Z/qZ O/I (O= integers in numberfield) 3

4 GAUSS SUMS IN PRIME FIELDS Theorem. (BK). For all ε > 0, there is δ > 0 such that if H < F p and H > pε, then max (a,p)=1 x H e p (ax) < C.p δ. H e p (x) = e 2πi p x Earlier results up to ε > 1 4 based on Stepanov s method 4

5 Conjecture. (MVW) H < F p, H log p H equidistributed Limitation of sum-product method log H > C log p loglog p 5

6 GAUSS SUMS IN Z/qZ Theorem. (B) For all ε > 0, there is δ > 0 such that if H < (Z/qZ) and H > q ε, then max (a,q)=1 x H e q (ax) < Cq δ H Estimate is uniform in q. 6

7 WEIL S INEQUALITY Theorem. (W) Let f(x) F p [x] be of degree d 1. Then 1 x p e p ( f(x) ) (d 1) p Problem. Obtain nontrivial estimates for d > p 7

8 MORDELL POLYNOMIALS Theorem. (B) f(x) = r i=1 a i x k i Z[X] (a i, p) = 1 (k i, p 1) < p 1 ε (1 i r) (k i k j, p 1) < p 1 ε (1 i j r) then p x=1 e p ( f(x) ) < Cp 1 δ where δ = δ(r, ε) > 0 8

9 VERSION FOR INCOMPLETE SUMS Theorem. Let θ 1,..., θ r F p satisfy ORD(θ i ) > p ε (1 i r) ORD(θ i θ 1 j ) > p ε (1 i j r) Let t > p ε. Then max a i F p t s=1 e p ( r i=1 a i θ s i ) < Cp δ t where δ = δ(r, ε) > 0 9

10 APPLICATIONS TO PSEUDO-RANDOMNESS AND CRYPTOGRAPHY Distribution of Diffie-Hellman triples {θ x, θ y, θ xy } Joint distribution of RSA sequences u n+1 = u e n (e = 2: Blum-Blum-Shub generator) 10

11 SCALAR SUM-PRODUCT THEOREMS PRODUCT THEOREMS IN MATRIX SPACE Theorem. (HELFGOTT) G = SL 2 (p) or SL 3 (p) Assume A G generates G and A < G 1 ε Then A.A.A > A 1+δ 11

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13 EXPANSION OF CAYLEY GRAPHS Theorem. (BG) Let S p = {g 1, g 1 1,..., g k, g 1 k } be a symmetric generating set for SL 2 (p), such that girth ( G(SL 2 (p), S p ) ) > τ log p Then the expansion coefficient of G ( SL 2 (p), S p ) admits a uniform lower bound c(τ) > 0 Problem. Remove large girth assumption (Relevant work by E. Breuillard) 12

14 G = GRAPH ON VERTEX SET V Definition. { X c(g) = inf X where X < 1 2 V } (Expansion coefficient of G) 13

15 LUBOTZKY WEISS CONJECTURE Let S be a finite subset of SL d (Z) generating a Zariski dense subgroup of SL d. Then there is q 0 Z such that the family of Cayley Graphs G ( SL d (Z/qZ), π q (S) ) with q Z +,(q, q 0 ) = 1 forms a family of expanders Also motivated by the work of B-Gamburd-Sarnak on prime sieving 14

16 CONNECTEDNESS OF THE GRAPH ROLE OF STRONG APPROXIMATION PROPERTY Theorem. Let G be a Zariski dense subgroup of SL d (Z). There is q 0 Z such that π q (G) = SL d (Z/qZ) if (q, q 0 ) = 1 π q : reduction mod q Matthews, Vaserstein, Weisfeiler, Pink 15

17 KNOWN RESULTS SL 2 (p) (B-G) SL 2 (q) (q square free) (B-G-S) SL 2 (p n ) (B-G) SL d (p n ) (p fixed prime) (B-G) (d arbitrary) SL 3 (p) 16

18 GENERAL OUTLINE OF PROOFS Expansion Property Spectral multiplicity argument (Sarnak Xue) Measure Convolution on SL(q) Non-commutative B-S-G theorem (Tao) Product Theorems in SL(q) Scalar Sum-Product Theorems Z/pZ, Z/qZ, O/I 17

19 PRIME SIEVING IN ORBITS OF LINEAR GROUPS USING EXPANSION PROPERTIES (B-G-S) 18

20 Λ SL 2 (Z) Λ nonelementary δ(λ) > 0 (arbitrary) Sieving in balls defined using either word-metric or Archimedian metric δ(λ) > 2 1 (Lax Phillips) δ(λ) 1 2 (Lalley Dolgopiat Naud) 19

21 Definition. r(z)= number of prime factors of z Z\{0} Theorem. (BGS) There is C(Λ) Z + such that for M {g = g 11 g 12 B g 21 g M ; r( g ij ) < C(Λ)} 22 B M (log M) 4 B M = {g Λ; ( g 2 ij ) 1/2 M} 20

22 Theorem. (BGS) Let f Q[x 1, x 2, x 3, x 4 ] taking integer values on Λ and not a multiple of g(x 1, x 2, x 3, x 4 ) = x 1 x 4 x 2 x 3 1 There is r = r(λ) Z + s.t. {x Λ f(x) has at most r prime factors} is Zariski dense in SL 2 21

23 EXPLICIT APPLICATIONS Example. Appolonian packings Appolonian packing corresponding to: Quadruple ( 6, 11, 14, 23) 22

24 DESCARTE FORM F(x 1, x 2, x 3, x 4 ) = 2(x x2 2 + x2 3 + x2 4 ) (x 1 + x 2 + x 3 + x 4 ) 2 O F = Orthogonal group A = S 1, S 2, S 3, S 4 = Appolonian packing group S 1 = S 2 = S 3 = S 4 =

25 CONJECTURE (BGS) (SL 2 (Z) analogue of Dirichlet s Theorem) Λ non-elementary subgroup of SL 2 (Z) b Z 2 primitive vector O = {gb g Λ} π(o) = {x O x 1, x 2 are prime} Then π(o) is Zariski dense in A 2 if no local obstruction: For every q 2, there is x O such that x 1 x 2 (Z/qZ) 24

26 SU(2) Theorem. (B-G) Let k 2 and g 1,..., g k algebraic elements in G = SU(2) generating a free group Consider the Hecke operator T : L 2 (G) L 2 (G) : Tf(x) = k j=1 Then there is a spectral gap λ 1 (T) < 2k γ ( f(gj x)+f(g 1 j x) ) where γ = γ(g 1,..., g k ) > 0 may be controlled by a noncommutative diophantine property 25

27 APPLICATIONS Banach-Ruziewicz problem Quantum-computation (Solovay-Kitaev algorithm) Orientations in Conway-Radin Quaquaversal tiling 26

28 THE QUAQUAVERSAL TILING 27

29 DISCRETIZED RING THEOREM Definition. A R, ε > 0 N(A, ε) = minimum number of ε-intervals covering A Theorem. Given 0 < δ 1, δ 2 < 1, there is δ 3 > 0 such that if A [1,2] satisfies (ε small) (i) N(A, ε) < ( 1ε ) 1 δ 1 (ii) N(A, ε 1 ) > ( 1ε1 ) δ 2 if ε < ε1 < 1 Then N(A+A, ε)+n(a.a, ε) > ( 1 ε ) δ 3N(A, ε) 29

30 ACTIONS OF NON-ABELIAN GROUPS ON TORI (BFLM) Margulis Furstenberg Guivarch Theorem. (BFLM) S = {g 1,..., g k } SL d (Z) S Zariski dense in SL d ν = 1 S g S δ g ξ T d irrational Then for ν ( ) -almost every sequence (x 1, x 2,...) the sequences x r x r 1... x 1 ξ and x 1 x 2... x r ξ (r ) are equidistributed in T d 30

31 Corollary. Let µ be a probability measure on T d which is ν-stationary, i.e. µ = µ ν = ν(g)g [µ] g then µ is a combination of Haar measure and atomic measure supported by rational points and µ is ν -invariant Remark. ν = SL d (Z): Furstenberg s stiffness problem Corollary. (Starkov, Muchnik, Guivarch) Let K T d be ν -invariant compact then K is finite or K = T d. 31

32 QUANTITATIVE EQUIDISTRIBUTION STATEMENT Theorem. (BFLM) There are constants c > 0 and C < such that if ξ T d \{0} and b Z d \{0}, b < e cn. Then either ( ) = or g ν(n) (g)e 2πi b,gξ ξ a < e cn q q < e 1 4 cn < e cn and ( ) < b C q c 32

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