The Erdos-Falconer distance problem in the tree setting

TL;DR

This paper extends recent advances in the Falconer distance problem to the setting of pinned trees within finite fields and Euclidean spaces, exploring the structure of distance sets in these generalized contexts.

Contribution

It introduces a general framework for analyzing distance problems in tree structures, building on recent work and extending results to broader configurations.

Findings

Established new bounds for distance sets in tree structures

Extended Falconer distance results to finite field and Euclidean tree settings

Provided a unified approach for pinned tree distance problems

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).