Full measure universality for Cantor Sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915686633177088 |
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| author | Shmerkin, Pablo Yavicoli, Alexia |
| author_facet | Shmerkin, Pablo Yavicoli, Alexia |
| contents | We investigate variants of the Erdős similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known fact that sets of positive Hausdorff dimension are not measure universal. We prove a weaker result for all Cantor sets $A$: there is a dense $G_δ$ set of full measure $X\subset\mathbb{R}^d$, such that for any bi-Lipschitz function $f:\mathbb{R}^d\to \mathbb{R}^d$, the set of translations $t$ such that $f(A)+t\subseteq X$ is of measure zero. Equivalently, there is a null set $B\subset\mathbb{R}^d$ such that $\mathbb{R}^d\setminus (f(A)+B)$ is null for all bi-Lipschitz functions $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Full measure universality for Cantor Sets Shmerkin, Pablo Yavicoli, Alexia Classical Analysis and ODEs 28A75, 28A78, 28A80 We investigate variants of the Erdős similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known fact that sets of positive Hausdorff dimension are not measure universal. We prove a weaker result for all Cantor sets $A$: there is a dense $G_δ$ set of full measure $X\subset\mathbb{R}^d$, such that for any bi-Lipschitz function $f:\mathbb{R}^d\to \mathbb{R}^d$, the set of translations $t$ such that $f(A)+t\subseteq X$ is of measure zero. Equivalently, there is a null set $B\subset\mathbb{R}^d$ such that $\mathbb{R}^d\setminus (f(A)+B)$ is null for all bi-Lipschitz functions $f$. |
| title | Full measure universality for Cantor Sets |
| topic | Classical Analysis and ODEs 28A75, 28A78, 28A80 |
| url | https://arxiv.org/abs/2503.21079 |