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arXiv · 2305.16026

Visible parts and slices of Ahlfors regular sets

Damian Dąbrowski
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Abstract

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 $s - \alpha(s-d+1),$ where $\alpha>0.183$ 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.

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Damian Dąbrowski. 2023-05-25. Visible parts and slices of Ahlfors regular sets. https://doi.org/10.19086/da.127570

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