New improvement to Falconer distance set problem in higher dimensions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913558507290624 |
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| author | Du, Xiumin Ou, Yumeng Ren, Kevin Zhang, Ruixiang |
| author_facet | Du, Xiumin Ou, Yumeng Ren, Kevin Zhang, Ruixiang |
| contents | We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04103 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New improvement to Falconer distance set problem in higher dimensions Du, Xiumin Ou, Yumeng Ren, Kevin Zhang, Ruixiang Classical Analysis and ODEs Combinatorics Metric Geometry 28A80, 28A78 We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$. |
| title | New improvement to Falconer distance set problem in higher dimensions |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A80, 28A78 |
| url | https://arxiv.org/abs/2309.04103 |