A two-parameter finite field Erdős-Falconer distance problem

Abstract

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