Intermediate dimensions of slices of compact sets
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915600913137664 |
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| author | Nicolas, Angelini Ursula, Molter |
| author_facet | Nicolas, Angelini Ursula, Molter |
| contents | $θ$ intermediate dimensions are a continuous family of dimensions that interpolate between Hausdorff and Box dimensions of fractal sets. In this paper we study the problem of the relationship between the dimension of a set $E\subset\mathbb{R}^d$ and the dimension of the slices $E\cap V$ from the point of view of intermediate dimensions. Here $V\in G(d,m)$, where $G(d,m)$ is the set of $m-$dimensional subspaces of $\mathbb{R}^d$. We obtain upper bounds analogous to those already known for Hausdorff dimension. In addition, we prove several corollaries referring to, among other things, the continuity of these dimensions at $θ=0$, a natural problem that arises when studying them. We also investigate which conditions are sufficient to obtain a lower bound that provides an equality for almost all slices. Finally, a new type of Frostman measures is introduced. These measures combine the results already known for intermediate dimensions and Frostman measures in the case of Hausdorff dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intermediate dimensions of slices of compact sets Nicolas, Angelini Ursula, Molter Classical Analysis and ODEs $θ$ intermediate dimensions are a continuous family of dimensions that interpolate between Hausdorff and Box dimensions of fractal sets. In this paper we study the problem of the relationship between the dimension of a set $E\subset\mathbb{R}^d$ and the dimension of the slices $E\cap V$ from the point of view of intermediate dimensions. Here $V\in G(d,m)$, where $G(d,m)$ is the set of $m-$dimensional subspaces of $\mathbb{R}^d$. We obtain upper bounds analogous to those already known for Hausdorff dimension. In addition, we prove several corollaries referring to, among other things, the continuity of these dimensions at $θ=0$, a natural problem that arises when studying them. We also investigate which conditions are sufficient to obtain a lower bound that provides an equality for almost all slices. Finally, a new type of Frostman measures is introduced. These measures combine the results already known for intermediate dimensions and Frostman measures in the case of Hausdorff dimension. |
| title | Intermediate dimensions of slices of compact sets |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.10376 |