Group actions and a multi-parameter Falconer distance problem

Abstract

In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given $\text{d}=(d_1,d_2,\dots ,d_{\ell })\in {\mathbb{N}}^{\ell }$ with $d_1+d_2+\cdots +d_{\ell }=d$ and $E\subseteq {\mathbb{R}}^d$, we define $${\Delta }_{\text{d}}(E)=\left\{\left(|x^{(1)}-y^{(1)}|,\dots ,|x^{(\ell )}-y^{(\ell )}|\right):x,y\in E\right\}\subseteq {\mathbb{R}}^{\ell },$$ where for $x\in {\mathbb{R}}^d$ we write $x=\left(x^{(1)},\dots ,x^{(\ell )}\right)$ with $x^{(i)}\in {\mathbb{R}}^{d_i}$. We ask how large does the Hausdorff dimension of $E$ need to be to ensure that the $\ell$-dimensional Lebesgue measure of ${\Delta }_{\text{d}}(E)$ is positive? We prove that if $2\le d_i$ for $1\le i\le \ell$, then the conclusion holds provided $$\dim (E)>d-\frac{\min d_i}{2}+\frac{1}{3}.$$ We also note that, by previous constructions, the conclusion does not in general hold if $$\dim (E)<d-\frac{\min d_i}{2}.$$ A group action derivation of a suitable Mattila integral plays an important role in the argument.