Additive energy and the Falconer distance problem in finite fields

TL;DR

This paper investigates the relationship between additive energy, Fourier decay, and the Falconer distance problem in finite fields, providing new bounds and conditions for the conjecture's validity in two dimensions.

Contribution

It introduces bounds based on additive energy and Fourier decay, and offers new conditions under which the Falconer distance conjecture holds in finite fields.

Findings

Lower bounds for vector sets based on additive energy

Conditions ensuring Falconer conjecture in two dimensions

Alternative proof for Salem sets satisfying the conjecture

Abstract

We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of the Fourier transform on given sets. As an application of our observation, we find sufficient conditions on sets where the Falconer distance conjecture for finite fields holds in two dimension. Moreover, we give an alternative proof of the theorem, due to Iosevich and Rudnev, that any Salem set satisfies the Falconer distance conjecture for finite fields.