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 $|A|\le p^{1/2}.$ 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).