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

Let $phi(x, y)colon mathbb{R}^dtimes mathbb{R}^dto mathbb{R}$ 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 $Esubset mathbb{R}^d$, $phi(E, E)$ 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 $mathbb{F}_q^d$.