Visible parts and slices of Ahlfors regular sets
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910751582584832 |
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| author | Dąbrowski, Damian |
| author_facet | Dąbrowski, Damian |
| contents | We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where $s>d-1$, we prove a much better estimate. In that case for almost every direction the Hausdorff dimension of the visible part is at most $s - α(s-d+1),$ where $α>0.183$ is absolute. The estimate is new even for self-similar sets satisfying the open set condition. Along the way, we prove a refinement of the Marstrand's slicing theorem for Ahlfors regular sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_16026 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Visible parts and slices of Ahlfors regular sets Dąbrowski, Damian Classical Analysis and ODEs 28A80 (primary) 28A78 (secondary) We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where $s>d-1$, we prove a much better estimate. In that case for almost every direction the Hausdorff dimension of the visible part is at most $s - α(s-d+1),$ where $α>0.183$ is absolute. The estimate is new even for self-similar sets satisfying the open set condition. Along the way, we prove a refinement of the Marstrand's slicing theorem for Ahlfors regular sets. |
| title | Visible parts and slices of Ahlfors regular sets |
| topic | Classical Analysis and ODEs 28A80 (primary) 28A78 (secondary) |
| url | https://arxiv.org/abs/2305.16026 |