Four-variable expanders over the prime fields

Let $\mathbb{F}_p$ be a prime field of order $p>2$, and $A$ be a set in $\mathbb{F}_p$ with very small size in terms of $p$. In this note, we show that the number of distinct cubic distances determined by points in $A\times A$ satisfies $$|(A-A)^3+(A-A)^3|\gg |A|^{8/7},$$ which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that $$\max \left\lbrace |A+A|, |f(A, A)|\right\rbrace\gg |A|^{6/5},$$ where $f(x, y)$ is a quadratic polynomial in $\mathbb{F}_p[x, y]$ that is not of the form $g(\alpha x+\beta y)$ for some univariate polynomial $g$.Comment: Accepted in PAMS, 201

A new sum-product estimate in prime fields

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.Comment: 16 page