Topological Erdős similarity conjecture and strong measure zero sets
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912235065966592 |
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| author | Jung, Yeonwook Lai, Chun-Kit |
| author_facet | Jung, Yeonwook Lai, Chun-Kit |
| contents | We resolve the topological version of the Erdős Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a result of the fact that the Borel conjecture is independent of the \textsf{ZFC} axiomatic set theory, the existence of an uncountable topologically universal set is independent of the \textsf{ZFC}. Moreover, our results can also be generalized to locally compact Polish groups ${\mathbb G}$. Returning to the measure side, we pose Full-Measure universal Erdős Similarity Conjecture with strongly meager sets via the duality of measure and category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological Erdős similarity conjecture and strong measure zero sets Jung, Yeonwook Lai, Chun-Kit Classical Analysis and ODEs Logic Metric Geometry We resolve the topological version of the Erdős Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a result of the fact that the Borel conjecture is independent of the \textsf{ZFC} axiomatic set theory, the existence of an uncountable topologically universal set is independent of the \textsf{ZFC}. Moreover, our results can also be generalized to locally compact Polish groups ${\mathbb G}$. Returning to the measure side, we pose Full-Measure universal Erdős Similarity Conjecture with strongly meager sets via the duality of measure and category. |
| title | Topological Erdős similarity conjecture and strong measure zero sets |
| topic | Classical Analysis and ODEs Logic Metric Geometry |
| url | https://arxiv.org/abs/2410.01275 |