On the Number of Dot Products Determined by a Large Set and One of its Translates in Finite Fields

TL;DR

This paper investigates the dot product structure of large sets in finite fields, proving that sufficiently large sets determine a substantial number of dot products and that their sum and difference sets cover the entire field.

Contribution

It establishes new bounds on the number of dot products determined by large sets in finite fields and shows the sum and difference sets generate the whole field.

Findings

Existence of pairs with large dot product sets

Sum and difference sets cover the entire finite field

Large sets determine many distinct dot products

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