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

Hunseok Kang, Doowon Koh, Dung The Tran

TL;DR

This work extends Raimi-type partition phenomena from the discrete setting to continuous compact groups, specifically the circle and the $n$-dimensional torus $\mathbb{T}^n$. It first solves the circle case with a constructive, measure-theoretic partition and a recursive rotation scheme, then lifts the result to higher dimensions via a slicing argument, using Fubini and product-structure of Haar measure. The main contributions are a circle partition with the Raimi rotation property and a torus version obtained by product-space reduction, complemented by discussion of polynomial extensions, quantitative questions, and connections to ergodic theory and distance problems. These results establish a robust continuous analogue of Raimi-type Ramsey phenomena and open several directions for further study in compact groups.

Abstract

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.

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

TL;DR

This work extends Raimi-type partition phenomena from the discrete setting to continuous compact groups, specifically the circle and the -dimensional torus . It first solves the circle case with a constructive, measure-theoretic partition and a recursive rotation scheme, then lifts the result to higher dimensions via a slicing argument, using Fubini and product-structure of Haar measure. The main contributions are a circle partition with the Raimi rotation property and a torus version obtained by product-space reduction, complemented by discussion of polynomial extensions, quantitative questions, and connections to ergodic theory and distance problems. These results establish a robust continuous analogue of Raimi-type Ramsey phenomena and open several directions for further study in compact groups.

Abstract

We extend Raimi's classical partition theorem to the continuous setting of the circle and -dimensional torus. Building on recent work of Hegyvári, Pach, and Pham in finite groups, we prove that there exist measurable partitions of the -dimensional torus 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.

Paper Structure

This paper contains 4 sections, 5 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.4
  3. Proof of Theorem 1.5
  4. Some remarks and open questions

Key Result

Theorem 1.1

Let $r, t, k, f\in \mathbb{N}$. Let $P^{(1)},\dots,P^{(f)}\in\mathbb{Z}[x]$ be non-constant polynomials with the properties that and the leading coefficients are positive. There exists a partition such that for any partition there exist $m\in\{1,\dots,t\}$, an element $x_0\in \mathbb{N}$, and a set $H\subset\mathbb{N}$ of positive lower density with the property that for every $h\in H$ and ever

Theorems & Definitions (10)

  • Theorem 1.1: Hegyvári--Pach--Pham HPP
  • Theorem 1.2: Hegyvári--Pach--Pham HPP
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:circle-raimi1']}
  • Claim 2.3
  • Theorem 3.1
  • proof