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arXiv · 2506.07251

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

Hunseok Kang·Doowon Koh·Firdavs Rakhmonov
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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 $\mathbb{F}_q$.

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Hunseok Kang, Doowon Koh, Firdavs Rakhmonov. 2025-06-08. The Erd\H{o}s-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields. https://arxiv.org/abs/2506.07251

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