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

Let $\phi(x, y)\colon \mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}$ be a function. We say $\phi$ 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 $E\subset \mathbb{R}^d$, $\phi(E, E)$ has a non-empty interior whenever $\dim_H(E)>\gamma$. The usual distance function, $\phi(x, y)=|x-y|$, 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 $\phi(E, E)=\mathbb{F}_q$ 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 $\mathbb{F}_q^d$.