Breaking the 6/5 threshold for sums and products modulo a prime

Abstract

Let $A\subset {\mathbb{F}}_p$ of size at most $p^{3/5}$. We show $$|A+A|+|AA|\gtrsim |A{|}^{6/5+c},$$ for $c=4/305$. Our main tools are the cartesian product point--line incidence theorem of Stevens and de Zeeuw and the theory of higher energies developed by the second author.