A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS

Abstract

We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if$A\subseteq \mathbb{F}_{p}$satisfies$|A|\leq p^{64/117}$then$\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018,arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy$E^{+}(P)$of some subset$P\subseteq A+A$. Our main novelty comes from reducing the estimation of$E^{+}(P)$to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’,Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.