Mattila–Sjölin Type Functions: A Finite Field Model
Journal
Vietnam Journal of Mathematics
Date Issued
2021
Author(s)
Abstract
Let ϕ(x, y) : ℝd× ℝd→ 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⊂ ℝd, ϕ(E,E) has a non-empty interior whenever dimH(E) > γ. The usual distance function, ϕ(x,y) = |x − y|, is conjectured to be a Mattila–Sjölin type function of index d2. In the setting of finite fields Fq, this definition is equivalent to the statement that ϕ(E, E) = Fq whenever |E|≫ qγ. The main purpose of this paper is to prove the existence of such functions with index d2 in the vector space Fqd. © 2021, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
Subjects
Erdős–Falconer distance problem; Falconer distance conjecture; Finite fields; Mattila–Sjölin type functions
Type
journal article