On the Mattila-Sj\"olin distance theorem for product sets

TL;DR

This paper improves the known threshold for the Mattila-Sj"olin distance theorem for product sets in higher dimensions, showing that the condition on the Hausdorff dimension can be lowered when the dimension is at least 5.

Contribution

The paper advances the Mattila-Sj"olin theorem by establishing a lower Hausdorff dimension threshold for the interior of the distance set in dimensions five and above.

Findings

Threshold for Hausdorff dimension improved for $d \,\geq\, 5$

Distance set $\,\Delta(E)$ has non-empty interior under weaker conditions

Results extend the applicability of the distance set theorem in higher dimensions

Abstract

Let $A$ be a compact set in $\mathbb{R}$, and $E=A^d\subset \mathbb{R}^d$. We know from the Mattila-Sj\"olin's theorem if $\dim_H(A)>\frac{d+1}{2d}$, then the distance set $\Delta(E)$ has non-empty interior. In this paper, we show that the threshold $\frac{d+1}{2d}$ can be improved whenever $d\ge 5$.