On the Mattila-Sjölin distance theorem for product sets

Abstract

Let $A$ be a compact set in $\mathbb{R}$, and $E=A^d\subset {\mathbb{R}}^d$. We know from the Mattila-Sjölin'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 $d\ge 5$.

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