aa r X i v : . [ m a t h . N T ] J un the sum-product estimate for largesubsets of prime fields M. Z. Garaev
Instituto de Matem´aticasUniversidad Nacional Aut´onoma de M´exicoCampus Morelia, Apartado Postal 61-3 (Xangari)C.P. 58089, Morelia, Michoac´an, M´exico [email protected]
Abstract
Let F p be the field of a prime order p. It is known that for anyinteger N ∈ [1 , p ] one can construct a subset A ⊂ F p with | A | = N such that max {| A + A | , | AA |} ≪ p / | A | / . In the present paper we prove that if A ⊂ F p with | A | > p / , thenmax {| A + A | , | AA |} ≫ p / | A | / . Key words: sum-product estimates, prime field, number of solutions
Let F p be the field of residue classes modulo a prime number p and let A ⊂ F p . Consider the sum set A + A = { a + b : a ∈ A, b ∈ A } AA = { ab : a ∈ A, b ∈ A } . From the work of Bourgain, Katz, Tao [2] and Bourgain, Glibichuk, Konya-gin [1] it is known that if | A | < p − δ , where δ > , then one has the sum-product estimate max {| A + A | , | AA |} ≫ | A | ε (1)for some ε = ε ( δ ) > . This result and its versions have found many impor-tant applications in various areas of mathematics.In the corresponding problem for integers (i.e., if the field F p is replacedby the set of integers) the conjecture of Erd¨os and Szemer´edi [4] is thatmax {| A + A | , | AA |} ≫ | A | − ε for any given ε > . At present the best knownbound in the integer problem is max {| A + A | , | AA |} ≫ | A | / (log | A | ) − / due to Solymosi [9].Explicit versions of (1) have been obtained in [5]–[8]. For subsets withrelatively small cardinalities (say, | A | < p / ), in [5] we proved thatmax {| A + A | , | AA |} ≫ | A | / (log | A | ) O (1) which was subsequently improved in [7] tomax {| A + A | , | AA |} ≫ | A | / (log | A | ) O (1) . In [5] we have also considered the case of subsets with larger cardinalities,which had been previously studied in [6]. We have shown, for example, thatmax {| A + A | , | AA |} ≫ min n | A | / p / , | A | / p − / o (log | A | ) O (1) . One may conjecture that the estimatemax {| A + A | , | AA |} ≫ min {| A | − ε , | A | / p / − ε } holds for all subsets A ⊂ F p . The motivation for the quantity | A | / p / − ε is clear from the construction in [3] which can be described as follows. Let g be a generator of F ∗ p . By the pigeon-hole principle, for any N ∈ [1 , p ] andfor any integer M ≈ p / N / (which we associate with M (mod p )), thereexists L such that |{ g x : 1 ≤ x ≤ M } ∩ { L + 1 , L + 2 , . . . , L + M }| ≫ M /p ≫ N. A ⊂ { g x : 1 ≤ x ≤ M } ∩ { L + 1 , L + 2 , . . . , L + M } with | A | ≈ N satisfies max {| A + A | , | AA |} ≪ p / | A | / . Thus, it follows that for any integer N ∈ [1 , p ] there exists a subset A ⊂ F p with | A | = N such thatmax {| A + A | , | AA |} ≪ p / | A | / . In the present paper we prove the following statement.
Theorem 1.
Let A ⊂ F p . Then | A + A || AA | ≫ min n p | A | , | A | p o . In view of the foregoing discussion, in the range | A | > p / our resultimplies the optimal in the general setting boundmax {| A + A | , | AA |} ≫ p / | A | / . The following generalization of Theorem 1 improves the correspondingresult from [11], where an analogy of the sum-product estimate from [6] hasbeen obtained for subsets of Z m , the ring of residue classes modulo m. Theorem 2.
Let A ⊂ Z m . Then | A + A || AA | ≫ min n m | A | , | A | m (cid:16)X d | md Acknowledgement. The author is grateful to S. V. Konyagin for veryuseful remarks. This work was supported by the Project PAPIIT IN 100307from UNAM. References [1] J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for thenumber of sums and products and for exponential sums in fields of primeorder, J. London Math. Soc. (2) (2006), 380–398.[2] J. Bourgain, N. Katz and T. Tao, A sum-product estimate in finite fieldsand their applications, Geom. Func. Anal. (2004), 27–57.[3] M.-C. Chang, Some problems in combinatorial number theory , Preprint.64] P. Erd¨os and E. Szemer´edi, On sums and products of integers. Studiesin pure mathematics , 213–218, Birkh˝auser, Basel,1983.[5] M. Z. Garaev, An explicit sum-product estimate in F p , Intern. Math.Res. Notices, to appear.[6] D. Hart, A. Iosevich and J. Solymosi, Sum product estimates in finitefields via Kloosterman sums, Intern. Math. Res. Notices, to appear.[7] N. H. Katz and Ch.-Y. Sun, A slight improvement to Garaev’s sumproduct estimate, Preprint.[8] N. H. Katz and Ch.-Y. Sun, Garaev’s inequality in fields not of primeorder, Preprint.[9] J. Solymosi, On the number of sums and products,