Sum-Product Problem: New Generalisations and Applications

Transcription

1 Sum-Product Problem: New Generalisations and Applications Igor E. Shparlinski Macquarie University ENS, Chaire France Telecom pour la sécurité des réseaux de télécommunications

2 1 Background Set Operations Let R be a ring. For m sets A 1,... A m R and a rational function we define the set F (X 1,..., X m ) R(X 1,..., X m ) F (A 1,..., A m ) = {F (a 1,...,a m ) is defined : We write k A and A (k) a 1 A 1,..., a m A m } for the sum and product of k copies of A, respectively.

3 2 Generic Question of Additive Combinatorics: Given several rational functions F 1,..., F n : Is at least one cardinality #F i (A 1,..., A m ) large? E.g., is max i #F i (A 1,..., A m ) substantially larger than max j #A j? Notation: and A B B A B = O(A) They allow more informative chains of relations like... now try A B = C A = O(B) = C

4 3 Sum-Product Problem The most studied case: A + A and A A Sets of Real Numbers: A IR Erdős & Szemerédi (1983) Proved: for some fixed δ > 0, max {# (A + A), # (A A)} (#A) 1+δ ; Conjectured: as #A max {# (A + A), # (A A)} (#A) 2+o(1). Solymosi (2009): (# (A + A)) 2 # (A A) (#A) 4+o(1) E.g., one can take any δ < 1/3, improving Nathanson (1997, 1/31), Ford (1998, 1/15), Elekes (1997, 1/4),...

5 4 Elekes & Ruzsa (2003): If # (A + A) (#A) then # (A A) (#A) 2+o(1) Open Question 1 What can we say about # (A + A) if # (A A) (#A)? Sets in Polynomial Rings Croot & Hart (2008): There is an absolute constant δ > 0 such that for all large sets A of monic polynomials over C # (A A) < (#A) 1+δ = # (A + A) (#A) 2.

6 5 Sets in Prime Fields: A IF p, p prime Bourgain, Katz & Tao (2003): For any fixed fixed ε > 0 there is δ > 0 such that if p ε #A p 1 ε, then max {# (A + A), # (A A)} (#A) 1+δ Bourgain, Glibichuk & Konyagin (2005): The lower bound #A p ε is not needed. Bourgain, Bukh, Garaev, Katz, Li, Shen, Tsimerman... ( ): Explicit versions: δ is an explicit function of ε; also for max {# (A + B), # (A B)}.

7 6 Current Status for max {# (A + A), # (A A)} max {# (A + A), # (A A)} (#A) 13/12, if #A p 1/2, (#A) 7/6 p 1/24+o(1), if p 1/2 #A p 35/68, (#A) 10/11 p 1/11+o(1), if p 35/68 #A p 13/20, (#A) 2 p 1/2, if p 13/20 #A p 2/3, (#A) 1/2 p 1/2, if #A p 2/3, The last bound is tight and this is the only range where a tight bound is known. Bourgain, Chang, Garaev,... ( ): Analogues for arbitrary finite fields and residue rings. Warning: Beware subfields/subrings.

8 7 Sum-Inversion Problem Sets of Real Numbers: A IR Elekes, Nathanson, Ruzsa (1999): # (A + A) # ( A ( 1) + A ( 1)) (#A) 5/2 Sets in Prime Fields: A IF p, p prime Bourgain (2006): For any ε > 0 there is δ > 0 such that if #A p 1 ε, then max { # (A + A), # ( A ( 1) + A ( 1))} (#A) 1+δ Chan & Shparlinski (2009): The method of Garaev (2007) + Bounds of Kloosterman sums: An explicit bound for #A p 1/2+ε.

9 8 Helfgott & Rudnev (2010): Explicit incidence theorem: Let P = A A IF 2 p be a set of points and let L(P) be the set of lines defined by the points from P. If #A < p 1/2 then #L(P ) (#A) 1+1/232. Together the argument of Bourgain (2006) this should almost instantly lead to an explicit lower bound on max { # (A + A), # ( A ( 1) + A ( 1))} for any A IF 2 p with #A < p 1 ε and in turn to explicit versions of some other results of Bourgain (2006). Has not been worked out yet.

10 9 Sum-Ratio Problem Sets of Real Numbers: A IR Li & Shen (2009): (# (A + A)) 2 # ( A A ( 1)) 1 4 (#A)4 Sets in Prime Fields: A IF p, p prime??? Probably Garaev s method should give something for #A p 1/2+ε

11 10 More Sets, More Operations Bourgain, Bukh, Chang, Croot, Garaev, Glibichuk, Hart, Katz, Konyagin, Li, Shen, Tsimerman ( ) A variety of results on and/or max {# (A A k ), # (A 1... A k )} max {# (A + B), #f(a, B)} Limitations of what can be true Let H be a multiplicative subgroup of IF p of order #H p 3/4+o(1). There exists m IF p such that for the set we have A = H {m + 1,..., m + p 3/4 } #A #Hp3/4 p However, for any integer k, p 1/2+o(1) max { # (k A), # ( A (k))} p 3/4+o(1).

12 11 Just One Set Sets of Real Numbers: A IR Elekes, Nathanson, Ruzsa (1999): # ( A + A ( 1)) (#A) 5/4 Garaev, Shen (2009): # (A (A + 1)) (#A) 5/4 Sets of Farey fractions of order Q Bourgain, Konyagin, Shparlinski (2008): For sets A, B of rational numbers with numerators and denominators bounded by Q #(A B) #A #B exp ( 9 log Q(log log Q) 1/2) Cilleruelo (2009): Improved the constant 9, showed that it is close to the best possible.

13 12 Open Question 2 Can we say anything interesting about #(A + B) where A, B are sets of rational numbers with numerators and denominators bounded by Q?... probably not as much as for #(A B) even if A and B are large: For and δ (0, 1) we can take q Q δ Then Since for A = B = { r qs : 1 r Q, 1 s Q/q #A = #B Q 2 δ r 1 qs 1 + r 2 qs 2 = r 1s 2 + r 2 s 1 qs 1 s 2 A + B we have r 1 s 2 + r 2 s 1 = O(Q 2 /q), we see that } #(A + B) Q2 q Q q Q q = Q4 q 3 #A #B Q δ

14 13 Sets in Prime Fields: A IF p, p prime Bourgain (2005): For any ε > 0 there is δ > 0 such that for #A p 1 ε # ( A + A ( 1)) (#A) 1+δ Garaev, Shen (2009): For A IF p with #A < p 1/2, # (A (A + 1)) (#A) 106/105+o(1). For any A IF p { # (A (A + 1)) min p#a, (#A)2 p 1/2 }. Glibichuk & Konyagin (2007) for A, B IF p with #A #B; Bourgain (2008) for any A, B IF p #(8 A B 8 A B) > 0.5 min {#A #B, p 1}

15 14 Applications Initial Applications Bourgain, Glibichuk & Konyagin (2005): Improving (the range of) Korobov (1972), Shparlinski (1991), Konyagin & Shparlinski (1998), Heath-Brown & Konyagin (1999), Konyagin (2002) on exponential sums over small subgroups of IF p Bourgain (2005): Improving (the range of) Mordell s bound on exponential sums with sparse polynomials; Canetti, Friedlander, Konyagin, Larsen, Lieman & Shparlinski (2001) on the distribution of the Diffie-Hellman triples (g x, g y, g xy ); Friedlander & Shparlinski (2001) on the distribution of the Blum-Blum-Shub pseudorandom number generator: Iterations of the power map x x e (mod m).

16 15 More Recent Applications Bourgain (2006): New bounds of short double Kloosterman sums a A b B exp(2πi(a + b) 1 /p) and improving Fouvry & Michel (1998) on exponential sums with reciprocals of primes 1/l l L l prime exp(2πil 1 /p) (based on the sum-inversion problem). Randomness extractors: Given a biased source of random bits obtain uniformly distributed bits.

17 16 Croot (2005), Bourgain (2006), Glibichuk (2006): Improving/generalising a result of Shparlinski (2002) on Erdős-Graham problem: Express every λ IF p as λ x x 1 k (mod p), 1 x 1,... x k p ε, with k k 0 (ε). Glibichuk (2006): k 0 (ε) = O(ε 2 ) improving k 0 (ε) = O(ε 3 ) obtained by Shparlinski (2002) using a different approach. Croot (2005), Bourgain (2006): Higher powers, systems of equations Shparlinski (2006): Distribution of values of the Ramanujan τ-function modulo p.

18 17 Very Recent Applications Bourgain, Ford, Konyagin & Shparlinski (2008): A bound on non-zero Fermat quotients: Define l(p) = min { a : (a p 1 1)/p 0 (mod p) }. Then l(p) (log p) 463/252+o(1) This improves Lenstra (1979), Granville (1990), Ihara (2005) who used different methods to get l(p) (log p) 2.

19 18 Bourgain, Konyagin, Pomerance & Shparlinski (2008): A bound on q g (x), the smallest x-pseudopower to base g. Those are numbers which look like powers of g modulo every p x but are not powers of g over Z. Then q g (x) exp( x) This improves Bach, Lukes, Shallit & Williams (1996) who derived q g (x) exp((1 + o(1))x) from the Chinese Remainder Theorem and the Prime Number Theorem

20 19 Cochrane & Pinner (2008): Applications of Glibichuk & Konyagin (2007) to the Waring problem modulo p: γ(k, p) = smallest integer s such that sums of kth powers of s integers represent every residue modulo p. If k 0 (mod (p 1)/2) then γ(k, p) 83k 1/2 The bound γ(k, p) k 1/2 is known as the Heilbronn conjecture. Cilleruelo, Ramana & Ramare (2009): Applications to the Sarközy problem about the gaps between consecutive elements in A A for A Z.

21 20 Balog & Broughan & Shparlinski (2009): #{x x 1 (mod p) : 1 x p} p 1/3+o(1). Elementary argument: For every d p 1, let X d be the set of solutions with gcd(x, p 1) = d (i) d x = #X d p/d; (ii) x x 1 (mod p) = x d 1 (mod p) = #X d d. Use (i) for d > p 1/2 and (ii) for d < p 1/2 = O(p 1/2+o(1) ) solutions.

22 21 Additional argument: # (X d X d ) 2p/d and # (X d X d ) d For d close to p 1/2 use the explicit bound of Bourgain & Garaev (2008) in the difference-product problem (with A A in place of A + A). More complicated argument: Balog & Broughan & Shparlinski (2009): For any a Z #{x x a (mod p) : 1 x p} p 12/23+o(1). The case of a a of large multiplicative order t p 12/23 is the bottleneck. The question has some cryptographic flavour (fixed points of the discrete logarithm map).

23 22 Chan & Shparlinski (2009): Bounds on the concentration function for solutions (x, y) of bivariate congruences. For any h, a nontrivial upper bound for the number of integer pairs in the square (x, y) [a, a + h] [b, b + h] which belong to: a modular exponential curve y g x (mod p) based on the sum-product results, thus the estimate is explicit; a modular hyperbola xy c (mod p) based on the sum-inversion results, thus the estimate is not explicit. Idea: Let Y be the set of y [b, b + h] such that y g x (mod p) with some x [a, a + h]. Clearly Y Y { h,..., h}; Y Y {g u : u {2a,..., 2a + 2h}} Use the explicit bound of Bourgain & Garaev (2008) in the difference-product problem.

24 23 Generalisations Sum-Product Problems on Elliptic Curves IE an elliptic curve over IF q Y 2 = X 3 + ax + b IE has a structure of an Abelian group (we use to denote the group operation). Variants: Shparlinski (2007): For sets A, B IE(IF q ) at least one of the sets and {x(p ) + x(q) : P A, Q B} {x(p Q) : P A, Q B} is large

25 24 Ahmadi & Shparlinski (2008): For P IE(IF q ) of order T, and sets A, B Z/T Z at least one of the sets (i) {x(ap ) + x(bp ) : a A, b B}, and {x(abp ) : a A, b B} (ii) {x(ap )x(bp ) : a A, b B}, and {x(abp ) : a A, b B}, is large

26 25 The above results are based on the idea of Garaev (2007) and are nontrivial only if max{#a, #B} q 1/2+η (or even larger in some cases). Open Question 3 Obtain nontrivial results below the square-root bound on the size.

27 26 Additive Combinatorics in Matrix Rings Noncommutativity hurts a lot..... A series of very interesting results by Bourgain & Gamburd, Chang, Helfgott,... ( ), however the progress is far behind the scalar case. Here is a concrete open problem which has a cryptographic motivation. It is posed by Maze, Monico & Rosenthal (2007) in the settings of matrices over semirings, however the IF q is a place to start. Let A, B, S be n n matrices over IF q. the set Consider M k (A, B, S) = {f(a)sg(b) : f, g IF q [X], deg f, g < k} One can assume that minimal polynomials of A and B are of degree n > k.

28 27 Clearly, we have the following trivial inequalities and #M k (A, B, S) q k #M k (A, A, A) q n. Open Question 4 Obtain a lower bound on the size of #M k (A, B, S) q (1+η)k with some fixed η > 0 for a wide class of matrices A, B and S. Some further conditions on A, B and S may also be necessary.

29 28 Links to Additive Combinatorics Let A = {f(a) : f IF q [X], deg f < k}, B = {Sg(B) : g IF q [X], deg g < k}. Clearly the cardinalities of sum sets #(A+A) = #A = q k and #(B+B) = #B = q k are small. Therefore one can expect that the cardinality of the product set #(A B) = #M k (A, B, S) is large. For sets A, B IF p, a similar statement is a very special case of a result of Bourgain (2005).