The Erd\"os-Falconer distance problem on the unit sphere in vector spaces over finite fields

TL;DR

This paper provides a graph-theoretic proof of the Erd"os-Falconer distance problem on the unit sphere in finite field vector spaces, originally proven using Fourier analysis, thereby offering a new perspective on the problem.

Contribution

It introduces a novel graph-theoretic approach to prove the Erd"os-Falconer distance conjecture on the unit sphere in finite fields.

Findings

Graph-theoretic proof of the Erd"os-Falconer distance conjecture

Validation of the conjecture for subsets of the unit sphere in finite fields

Alternative proof method to Fourier analysis

Abstract

art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erd\"os-Falconer distance conjecture holds for subsets of the unit sphere in $\mathbbm{F}_q^d$. In this note, we give a graph theoretic proof of this result.