Hausdorff dimension and countable Borel equivalence relations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909370517815296 |
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| author | Marks, Andrew Rossegger, Dino Slaman, Theodore |
| author_facet | Marks, Andrew Rossegger, Dino Slaman, Theodore |
| contents | We show that if $E$ is a countable Borel equivalence relation on $\mathbb{R}^n$, then there is a closed subset $A \subset [0,1]^n$ of Hausdorff dimension $n$ so that $E \restriction A$ is smooth. More generally, if $\leq_Q$ is a locally countable Borel quasi-order on $2^ω$ and $g$ is any gauge function of lower order than the identity, then there is a closed set $A$ so that $A$ is an antichain in $\leq_Q$ and $H^g(A) > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension and countable Borel equivalence relations Marks, Andrew Rossegger, Dino Slaman, Theodore Logic 03E15, 28A78 We show that if $E$ is a countable Borel equivalence relation on $\mathbb{R}^n$, then there is a closed subset $A \subset [0,1]^n$ of Hausdorff dimension $n$ so that $E \restriction A$ is smooth. More generally, if $\leq_Q$ is a locally countable Borel quasi-order on $2^ω$ and $g$ is any gauge function of lower order than the identity, then there is a closed set $A$ so that $A$ is an antichain in $\leq_Q$ and $H^g(A) > 0$. |
| title | Hausdorff dimension and countable Borel equivalence relations |
| topic | Logic 03E15, 28A78 |
| url | https://arxiv.org/abs/2410.22034 |