Slightly improved sum-product estimates in fields of prime order

Abstract

Abstract. Let Fp be the field of residue classes modulo a prime number p and let A be a nonempty subset of Fp. In this paper we show that if |A| p 0.5 , then max{|A ± A|, |AA|} |A| 13/12 ;These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered. IntroductionLet F p be the field of residue classes modulo a prime number p and let A, B be two nonempty subsets of F p . Define the sum set, difference set and product set of A and B respectively byFrom the work of Bourgain, Katz, Tao [4] and Bourgain, Glibichuk, Konyagin [3], it is known that if |A| p δ (see Section 2 for the definitions of , , , and ∼), where δ < 1, then one has the sum-product estimate max{|A + A|, |AA|} |A| 1+ǫ for some ǫ = ǫ(δ) > 0. These kinds of results have found many important applications in various areas of mathematics, and people want to know some quantitative relationships between δ and ǫ in certain ranges of |A|. For the case |A| p 0.5 , the pioneer work was due to Hart, Iosevich and Solymosi [12] via Kloosterman sums. See [6,8,18] for further improvements. Note also all lower bounds in [6,8,12,18]

An improved sum-product estimate for general finite fields

“…Since Garaev's sum-product estimate [3], there have been a number of small improvements made courtesy of more subtle arguments by Katz-Shen [5], Bourgain-Garaev [1], Shen [9], Li [6] and most recently Rudnev [7]; so the following result of Rudnev represents the state of the art.…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…For smaller sets the situation is less understood and is far from being settled. Various improvements of the initial result of [36] are due to Bourgain & Garaev [29], Garaev [95], Hart, Iosevich & Solymosi [118], Katz & Shen [141], Li [155] and Rudnev [165] and many others. The current state of affairs has been conveniently summarized by Bukh & Tsimerman [49] which we now reproduce here.…”

“…The current state of affairs has been conveniently summarized by Bukh & Tsimerman [49] which we now reproduce here. The bounds of Theorem 3.2 are due to the work of Garaev [96] (the last two bounds), Li [155] (the second and third bounds) and Rudnev [165] (the first bound). Rudnev [165,Remark 2] also mentions the bound…”

“…Since that time, this area has enjoyed a lot of attention, has been developed in various directions and has had a wide range of important applications, see [15,29,30,35,96,98,119,155,165,169,172] and references therein. Here we describe a series of recently emerged directions and present some concrete results.…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…See also [56,127,128,201,202,214,282,284] for other generalizations and improvements of Garaev's result. As an application, Shparlinski [290] using Rudnev's result [266], estimates the cardinality, #Γ p (T ), of the set…”