On the multiparameter Falconer distance problem

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 $d-\frac {\operatorname {min}(d_i)}{2}+\frac {1}{4}$ if $\operatorname {min}(d_i)$ is even, $d-\frac {\operatorname {min}(d_i)}{2}+\frac {1}{4} +\frac {1}{4\operatorname {min}(d_i)}$ if $\operatorname {min}(d_i)$ 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.