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, \( (E + E) \cdot (E – E) = \mathbb{F}_q.\)