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arXiv · 2512.00935

Raimi's theorem for the $n$-dimensional torus

Hunseok Kang·Doowon Koh·Dung The Tran
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Abstract

We extend Raimi's classical partition theorem to the continuous setting of the circle and $n$-dimensional torus. Building on recent work of Hegyv\'ari, 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.

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BibTeXRIS

Hunseok Kang, Doowon Koh, Dung The Tran. 2025-11-30. Raimi's theorem for the $n$-dimensional torus. https://arxiv.org/abs/2512.00935

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