The Erdos-Falconer distance problem in the tree setting

Abstract

The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set $\Delta(A)$ has positive Lebesgue measure. In a very recent paper, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for $E\subset\mathbb{F}_p^2$ with $|E|\gg p^{5/4}$, there exist many points $x\in E$ such that the number of distinct distances from $x$ is at least $cp$. The main purpose of this paper is to provide extensions in a very general structure of pinned trees, which is inspired by the recent work due to Ou and Taylor (2021).