A New Sum–product Estimate in Prime Fields

Abstract

In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if A ⊆ F p satisfies |A| p 64/117 then max{|A ± A|, |AA|} |A| 39/32 .Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy E + (P ) of some subset P ⊆ A + A. Our main novelty comes from reducing the estimation of E + (P ) to a point-plane incidence bound of Rudnev rather than a point line incidence bound of Stevens and de Zeeuw as done

On products of shifts in arbitrary fields

“…The 6/5 threshold has recently been broken, see [1], [2], and [3]. The following theorem was proved in [2] by Rudnev, Shakan, and Shkredov, and is the current state of the art bound.…”

Attaining the exponent $5/4$ for the sum-product problem in finite fields

“…Their result was improved by Chen, Kerr and Mohammadi [3] through a more efficient application of these techniques. Rudnev, Shakan and Shkredov [16] further advanced the record by developing a new double-counting argument, which remains present in this paper, to yield the current stateof-the-art.…”