On the multiparameter Falconer distance problem

Abstract

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell \ge 1$ and ${\mathbb{R}}^d={\mathbb{R}}^{d_1}\times \cdots \times {\mathbb{R}}^{d_{\ell }}$, $d_i\ge 2$. 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.

31 pages; final version to appear in Transactions of the AMS

References in corpus (4)

• On the dimension and smoothness of radial projections
• A sharp exceptional set estimate for visibility
• A two-parameter finite field Erdős-Falconer distance problem
• Group actions and a multi-parameter Falconer distance problem