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arXiv · 2410.22034

Hausdorff dimension and countable Borel equivalence relations

Andrew Marks·Dino Rossegger·Theodore Slaman
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Abstract

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 $2^{\omega}$ 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 $H^g(A) > 0$.

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Andrew Marks, Dino Rossegger, Theodore Slaman. 2024-10-29. Hausdorff dimension and countable Borel equivalence relations. https://arxiv.org/abs/2410.22034

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