Mattila–Sjölin Type Functions: A Finite Field Model

Abstract

Let \(\phi (x, y)\colon \mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a function. We say ϕ is a Mattila–Sjölin type function of index γ if γ is the smallest number satisfying the property that for any compact set \(E\subset \mathbb {R}^{d}\), ϕ(E,E) has a non-empty interior whenever \(\dim _{H}(E)>\gamma \). The usual distance function, ϕ(x,y) = |x − y|, is conjectured to be a Mattila–Sjölin 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|≫ qγ. The main purpose of this paper is to prove the existence of such functions with index \(\frac {d}{2}\) in the vector space \(\mathbb {F}_{q}^{d}\).