On GILP's group theoretic approach to Falconer's distance problem

TL;DR

This paper extends a group-theoretic approach to Falconer's distance problem, removing technical conditions and applying it to finite point configurations and a Fourier-free method, connecting incidence estimates to the Kakeya problem.

Contribution

It removes a continuity condition in GILP's theorem and extends the dimension bounds for finite point configurations in Falconer's problem.

Findings

Extended GILP's method to finite configurations with new bounds.

Introduced a Fourier-free approach using tubular incidence estimates.

Linked incidence estimates to the Kakeya problem.

Abstract

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf-Iosevich-Liu-Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP's theorem in [GILP15]. This allows us to extend the Wolff-Erdogan dimension bound for distance sets to finite points configurations with $k$ points for $k\in\{2,\dots,n+1\}.$ At the end of this paper, we extend this group-theoretic method and illustrate a `Fourier free' approach to Falconer's distance set problem for the Lebesgue measure. We explain how to use tubular incidence estimates in distance set problems. Curiously, tubular incidence estimates are also related to the Kakeya problem.