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.

Locations

• Bulletin of the Australian Mathematical Society - View
• arXiv (Cornell University) - View - PDF