Mattila–Sjölin Type Functions: A Finite Field Model

Let ϕ ( x , y ) : ℝ d × ℝ d → 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 ⊂ ℝ d , ϕ ( E , E ) has a non-empty interior whenever dim H ( E ) > γ . The usual distance function, ϕ ( x , y ) = | x − y |, is conjectured to be a Mattila–Sjölin type function of index d 2 . In the setting of finite fields F q , this definition is equivalent to the statement that ϕ ( E , E ) = F q whenever | E |≫ q γ . The main purpose of this paper is to prove the existence of such functions with index d 2 in the vector space F q d .