The Erdős-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields

Abstract

In this paper, we study the cardinality of the distance set $\hat{I}”(A,B)$ 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 $|\hat{I}”(A,B)|>q/2$, where $|\hat{I}”(A,B)|$ 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 Erdős-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 ${\mathbb{F}}_q$.

13 pages