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arXiv · 2103.11418

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

Doowon Koh·Thang Pham·Chun-Yen Shen
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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 $d\ge 5$.

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Doowon Koh, Thang Pham, Chun-Yen Shen. 2021-03-21. On the Mattila-Sj\"olin distance theorem for product sets. https://arxiv.org/abs/2103.11418

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