Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.15709 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917963152490496 |
|---|---|
| author | Borges, Tainara Foster, Benjamin Ou, Yumeng Palsson, Eyvindur |
| author_facet | Borges, Tainara Foster, Benjamin Ou, Yumeng Palsson, Eyvindur |
| contents | For a compact set $E\subset\mathbb{R}^d$, $d\geq 2$, consider the pinned distance set $Δ^{y}(E)=\lbrace |x-y| : x\in E\rbrace$. 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 $Δ^{y}(E)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonempty interior of pinned distance and tree sets Borges, Tainara Foster, Benjamin Ou, Yumeng Palsson, Eyvindur Classical Analysis and ODEs For a compact set $E\subset\mathbb{R}^d$, $d\geq 2$, consider the pinned distance set $Δ^{y}(E)=\lbrace |x-y| : x\in E\rbrace$. 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 $Δ^{y}(E)$ 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. |
| title | Nonempty interior of pinned distance and tree sets |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.15709 |