A two-parameter finite field Erd\H{o}s-Falconer distance problem

TL;DR

This paper investigates a two-parameter finite field Erdős-Falconer distance problem, establishing conditions under which the distance set covers the entire finite field product and providing sharp bounds for specific cases.

Contribution

It introduces a new two-parameter variant of the Erdős-Falconer problem over finite fields and proves sharp bounds for the size of the distance set in this setting.

Findings

If |E||F| ≥ C q^{k+2l+1}, then B_{k,l}(E,F) equals the entire product space.

The result is sharp when k is odd.

For l=k=2 and q ≡ 3 mod 4, a lower bound on |E||F| ensures a large distance set.

Abstract

We study the following two-parameter variant of the Erd\H os-Falconer distance problem. Given $E,F \subset {\Bbb F}_q^{k+l}$, $l \geq k \ge 2$, the $k+l$-dimensional vector space over the finite field with $q$ elements, let $B_{k,l}(E,F)$ be given by $$\{(\Vert x'-y'\Vert, \Vert x"-y" \Vert): x=(x',x") \in E, y=(y',y") \in F; x',y' \in {\Bbb F}_q^k, x",y" \in {\Bbb F}_q^l \}.$$ We prove that if $|E||F| \geq C q^{k+2l+1}$, then $B_{k,l}(E,F)={\Bbb F}_q \times {\Bbb F}_q$. Furthermore this result is sharp if $k$ is odd. For the case of $l=k=2$ and $q$ a prime with $q \equiv 3 \mod 4$ we get that for every positive $C$ there is $c$ such that $$ \text{if } |E||F|>C q^{6+\frac{2}{3}}\text{, then } |B_{2,2}(E,F)|> c q^{2}.$$