Mattila--Sj\"{o}lin type functions: A finite field model

Let $\varphi (x,y)\text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}$ be a function. We say $\varphi$ is a Mattila–Sj"{o}lin type function of index $\gamma$ if $\gamma$ is the smallest number satisfying the property that for any compact set $E\subset {\mathbb{R}}^d$, $\varphi (E,E)$ has a non-empty interior whenever ${\dim }_H(E)>\gamma$. The usual distance function, $\varphi (x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, is conjectured to be a Mattila–Sj"{o}lin type function of index $\frac{d}{2}$. In the setting of finite fields ${\mathbb{F}}_q$, this definition is equivalent to the statement that $\varphi (E,E)={\mathbb{F}}_q$ whenever $|E|\gg q^{\gamma}$. The main purpose of this paper is to prove the existence of such functions with index $\frac{d}{2}$ in the vector space ${\mathbb{F}}_q^d$.