A note on conditional expanders over prime fields

Abstract

Let $\mathbb{F}_p$ be a prime field of order $p,$ and $A$ be a set in $\mathbb{F}_p$ with $|A| \leq p^{1/2}.$ In this note, we show that \[\max\{|A+A|, |f(A, A)|\}\gtrsim |A|^{\frac{6}{5}+\frac{4}{305}},\] where $f(x, y)$ is a non-degenerate quadratic polynomial in $\mathbb{F}_p[x, y].$ This improves a recent result given by Koh, Mojarrad, Pham, Valculescu (2018).