On the sum of the squared multiplicities of the distances in a point set over finite fields

TL;DR

This paper investigates a finite field version of Erdős's conjecture concerning the sum of squared distance multiplicities in point sets, utilizing spectral graph theory to estimate hinge counts.

Contribution

It introduces a novel finite field analog of Erdős's conjecture and applies spectral graph techniques to estimate distance multiplicity sums.

Findings

Established bounds on the sum of squared distance multiplicities

Connected hinge counts in spectral graphs to distance problems

Provided new insights into finite field geometric configurations

Abstract

We study a finite analog of a conjecture of Erd\"os on the sum of the squared multiplicities of the distances determined by an $n$-element point set. Our result is based on an estimate of the number of hinges in spectral graphs.