Raimi's theorem for the $n$-dimensional torus
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914175009161216 |
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| author | Kang, Hunseok Koh, Doowon Tran, Dung The |
| author_facet | Kang, Hunseok Koh, Doowon Tran, Dung The |
| contents | We extend Raimi's classical partition theorem to the continuous setting of the circle and $n$-dimensional torus. Building on recent work of Hegyvári, Pach, and Pham in finite groups, we prove that there exist measurable partitions of the $n$-dimensional torus $\mathbb{T}^n$ with the property that for any finite measurable cover, some translated part of the cover has positive measure intersection with every partition element. Our proof adapts combinatorial arguments from the finite setting using measure-theoretic techniques and slicing arguments in product spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Raimi's theorem for the $n$-dimensional torus Kang, Hunseok Koh, Doowon Tran, Dung The Combinatorics Classical Analysis and ODEs Number Theory We extend Raimi's classical partition theorem to the continuous setting of the circle and $n$-dimensional torus. Building on recent work of Hegyvári, Pach, and Pham in finite groups, we prove that there exist measurable partitions of the $n$-dimensional torus $\mathbb{T}^n$ with the property that for any finite measurable cover, some translated part of the cover has positive measure intersection with every partition element. Our proof adapts combinatorial arguments from the finite setting using measure-theoretic techniques and slicing arguments in product spaces. |
| title | Raimi's theorem for the $n$-dimensional torus |
| topic | Combinatorics Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2512.00935 |