In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${ \textbf{ d}}=(d_1,d_2, \dots, d_{ \ell})\in \mathbb{ N}^{ \ell}$ with $d_1+d_2+\dots+d_{ \ell}=d$ and $E \subseteq \mathbb{ R}^d$, we define $$ \Delta_{ { \textbf{ d}}}(E) = \left\{ \left(|x^{ (1)}-y^{ (1)}|,\ldots,|x^{ (\ell)}-y^{ (\ell)}|\right) : x,y \in E \right\} \subseteq \mathbb{ R}^{ \ell}, $$ 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 $\Delta_{ { \textbf{ d}}}(E)$ is positive? We prove that if $2 \leq d_i$ for $1 \leq i \leq \ell$, 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.
Categories: math.CA