On the multiparameter Falconer distance problem

TL;DR

This paper extends the Falconer distance problem to a multiparameter setting, establishing conditions under which the distance set of a fractal set has positive measure, using a new radial projection theorem.

Contribution

It introduces a multiparameter radial projection theorem and proves that certain fractal sets have distance sets of positive measure in a multiparameter context.

Findings

Sets with Hausdorff dimension above a specific threshold have positive measure distance sets.

A new multiparameter radial projection theorem for fractal measures is developed.

The results generalize classical Falconer distance problem to a multiparameter framework.

Abstract

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$ with Hausdorff dimension larger than $d-\frac{\min(d_i)}{2}+\frac{1}{4}$ if $\min(d_i) $ is even, $d-\frac{\min(d_i)}{2}+\frac{1}{4}+\frac{1}{4\min(d_i)}$ if $\min(d_i) $ is odd, we prove that the multiparameter distance set of $E$ has positive $\ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.