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arXiv · 2503.15709

Nonempty interior of pinned distance and tree sets

Tainara Borges·Benjamin Foster·Yumeng Ou·Eyvindur Palsson
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Abstract

For a compact set $E\subset\mathbb{R}^d$, $d\geq 2$, consider the pinned distance set $\Delta^{y}(E)=\lbrace |x-y| : x\in E\rbrace$. Peres and Schlag showed that if the Hausdorff dimension of $E$ is bigger than $\frac{d+2}{2}$ with $d\geq 3$, then there exists a point $y\in E$ such that $\Delta^{y}(E)$ has nonempty interior. In this paper we obtain the first non-trivial threshold for this problem in the plane, improving on the Peres--Schlag threshold when $d=3$, and we extend the results to trees using a novel induction argument.

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Tainara Borges, Benjamin Foster, Yumeng Ou, Eyvindur Palsson. 2025-03-19. Nonempty interior of pinned distance and tree sets. https://arxiv.org/abs/2503.15709

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