Group actions, the Mattila integral and applications

TL;DR

This paper introduces a simplified generalized Mattila integral and applies it to prove new results on the measure of product distance sets and dot product sets in Euclidean spaces, advancing geometric measure theory.

Contribution

It develops a simplified generalized Mattila integral and applies it to establish measure positivity results for product distances and dot product sets.

Findings

Product of distances set has positive measure if imensionality exceeds a threshold.

Dot product set has positive measure when the sum of dimensions exceeds 4.

Introduces a new, simpler approach to the Mattila integral.

Abstract

The Mattila integral, $$ {\mathcal M}(\mu)=\int {\left( \int_{S^{d-1}} {|\widehat{\mu}(r \omega)|}^2 d\omega \right)}^2 r^{d-1} dr,$$ developed by Mattila, is the main tool in the study of the Falconer distance problem. In this paper, with a very simple argument, we develop a generalized version of the Mattila integral. Our first application is to consider the product of distances $$(\Delta(E))^k= \left\{\prod_{j=1}^k |x^j-y^j|: x^j, y^j\in E\right\} $$ and show that when $d\geq 2$, $(\Delta(E))^k$ has positive Lebesgue measure if $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4k-1}$. Another application is, we prove for any $E,F,H\subset\mathbb{R}^2$, $\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)+\dim_{\mathcal{H}}(H)>4$, the set $$E\cdot(F+H)=\{x\cdot(y+z): x\in E, y\in F, z\in H\}$$ has positive Lebesgue.