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 $A\subseteq {\mathbb{F}}_p$ satisfies $|A|\le p^{64/117}$ then $$\max \{|A\pm A|,|AA|\}\gtrsim |A{|}^{39/32}.$$ Our argument builds on and improves some recent results of Shakan and Shkredov 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 rather than a point line incidence bound of Stevens and de Zeeuw as done by Shakan and Shkredov.

16 pages