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

TL;DR

This paper generalizes Raimi's partition theorem to the continuous setting of the n-dimensional torus, demonstrating the existence of measurable partitions with a universal intersection property for finite covers.

Contribution

It extends Raimi's theorem to the n-dimensional torus using measure-theoretic and combinatorial techniques, bridging finite group results with continuous spaces.

Findings

Existence of measurable partitions with universal intersection properties.

Extension of Raimi's theorem to higher-dimensional tori.

Application of measure-theoretic and slicing techniques.

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.