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

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 $$\{(\mathrm{\parallel }{x}^{\mathrm{\prime }}-y^{\mathrm{\prime }}\mathrm{\parallel },\mathrm{\parallel }x\mathrm{"}-y\mathrm{"}\mathrm{\parallel }):x=(x^{\mathrm{\prime }},x\mathrm{"})\in E,y=(y^{\mathrm{\prime }},y\mathrm{"})\in F;x^{\mathrm{\prime }},y^{\mathrm{\prime }}\in {\mathbb{F}}_q^k,x\mathrm{"},y\mathrm{"}\in {\mathbb{F}}_q^l\}\mathrm{.}$$ We prove that if $\mathrm{\mid }E\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }\ge C{q}^{k+2l+1}$, then $B_{k,l}(E,F)={\Bbb F}_q \times {\Bbb F}_q$\mathrm{.}Furthermorethisresultissharpif$k$ is odd. For the case of $l=k=2$ and $q$ a prime with $q\equiv 3\mathrm{m}\mathrm{o}\mathrm{d}4$ we get that for every positive $C$ there is $c$ such that $$\text{if }\mathrm{\mid }E\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }>C{q}^{6+\frac{2}{3}}\text{, then }\mathrm{\mid }{B}_{2,2}(E,F)\mathrm{\mid }>c{q}^2\mathrm{.}$$