On the number of dot products determined by a large set and one of its translates in finite fields

Abstract

Let $E\subseteq {\mathbb{F}}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements, which satisfies $|E|>q$. There exist $x,y\in E$ such that $|E\cdot (y–x)|>q/2$. In particular, $(E+E)\cdot (E–E)={\mathbb{F}}_q.$