The Erdos-Falconer distance problem in the tree setting

Abstract

The recent breakthrough result of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set A ⊂ R 2 , if the Hausdorff dimension of A is greater than 5 4 , then the distance set ∆(A) has positive Lebesgue measure. Ou and Taylor (2021) recently generalized this result to the setting of (distance) trees. The main purpose of this paper is to study the discrete version of their result over both prime fields and arbitrary fields for small sets.