The Erd\H{o}s-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields

TL;DR

This paper investigates the number of distinct distances between two sets in finite fields, especially when one set is contained in a coordinate plane, and establishes thresholds for when the distance set is large, improving understanding of the Erdős-Falconer problem.

Contribution

It proves a new lower bound on the size of the distance set between arbitrary sets and $k$-coordinatable sets in finite fields, extending known thresholds and applications.

Findings

Established that if |A||B| > 2q^d, then the distance set size exceeds q/2.

Recovered the sharp (d+1)/2 threshold for the Erdős-Falconer problem in odd dimensions.

Provided an improved result on the Box distance problem for cases where 2 is a square in the finite field.

Abstract

In this paper, we study the cardinality of the distance set $\Delta(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane up to translations and rotations, we prove that if $|A||B| > 2q^d$, then $|\Delta(A, B)| > q/2$, where $|\Delta(A, B)|$ denotes the number of distinct distances between elements of $A$ and $B$. In particular, we show that our result recovers the sharp $(d+1)/2$ threshold for the Erd\H{o}s-Falconer distance problem in odd dimensions, where distances are determined by a single set. As an application, we also obtain an improved result on the Box distance problem posed by Borges, Iosevich, and Ou, in the case where $2$ is a square in $\mathbb{F}_q$.