# Distribution of distances in positive characteristic

## TL;DR

This paper improves bounds on the distribution of distances in finite field point sets with Cartesian product structure, using incidence bounds to show more distances are covered and pair counts are near expected values.

## Contribution

It introduces new results that break the previous exponent barrier for distance coverage in prime fields with Cartesian product sets, leveraging recent incidence bounds.

## Key findings

- Coverage of all distances with fewer points in Cartesian product sets.
- Number of point pairs at each distance is close to the expected value.
- Improved bounds using Rudnev's point-plane incidence theorem.

## Abstract

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\mathcal{E}|\ge 4q^{\frac{d+1}{2}}$, then $\Delta(\mathcal{E})=\mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $\mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover all distances. We also show that the number of pairs of points in $\mathcal{E}$ of any given distance is close to its expected value.