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Polynomial extensions of Raimi's theorem

Norbert Hegyvari, Janos Pach, Thang Pham

TL;DR

This work extends Raimi's partition-unavoidability from N to higher-dimensional lattices and further to polynomial shifts in $\mathbb{N}^k$ under the guiding influence of Weyl equidistribution. By combining diagonal shifts, polynomial patterns, and duality theory, the authors construct partitions $\bigcup_{i=1}^r E_i$ such that every finite coloring yields a single color class $F_m$ with shifts of the form $x_0+P^{(j)}(h)$ aligning with all partition pieces for infinitely many $h$. The main novelty lies in creating a positive-density set of shifts that simultaneously accommodate all polynomials $P^{(j)}$, accomplished via a careful analysis of a relation lattice, a subtorus $H$, and equidistribution on $H$, together with finite-structure analogues for abelian groups and $SL_2(\mathbb{F}_q)$. These results reveal robustness of Raimi-type intersection properties under polynomial patterns and extend the toolkit for partition regularity to new combinatorial and algebraic settings. The finite-group extensions demonstrate the breadth of the phenomenon beyond the infinite lattice, with potential implications for incidence and expansion problems in finite fields.

Abstract

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after an appropriate shift (translation). We establish a polynomial extension of this result, proving that such intersections persist under polynomial shifts in any dimension. Let $P^{(1)},\dots,P^{(f)}\in\mathbb{Z}[x]$ be non-constant polynomials with positive leading coefficients and $P^{(j)}(0)=0$ for every $j$. We construct a partition of $\mathbb{N}^k$ into an arbitrarily fixed finite number of pieces such that for any coloring of $\mathbb{N}^k$ with finitely many colors, there exist $x_0\in \mathbb{N}$ and a single color class that meets all partition pieces after shifts by $x_0+P^{(j)}(h)$ in each of the $k$ coordinate directions, for every $j$ and infinitely many values $h\in \mathbb{N}$. Our proof exploits Weyl's equidistribution theory, Pontryagin duality, and the structure of polynomial relation lattices. We also prove some finite analogues of the above results for abelian groups and $SL_2(\mathbb{F}_q)$.

Polynomial extensions of Raimi's theorem

TL;DR

This work extends Raimi's partition-unavoidability from N to higher-dimensional lattices and further to polynomial shifts in under the guiding influence of Weyl equidistribution. By combining diagonal shifts, polynomial patterns, and duality theory, the authors construct partitions such that every finite coloring yields a single color class with shifts of the form aligning with all partition pieces for infinitely many . The main novelty lies in creating a positive-density set of shifts that simultaneously accommodate all polynomials , accomplished via a careful analysis of a relation lattice, a subtorus , and equidistribution on , together with finite-structure analogues for abelian groups and . These results reveal robustness of Raimi-type intersection properties under polynomial patterns and extend the toolkit for partition regularity to new combinatorial and algebraic settings. The finite-group extensions demonstrate the breadth of the phenomenon beyond the infinite lattice, with potential implications for incidence and expansion problems in finite fields.

Abstract

Raimi's theorem guarantees the existence of a partition of into two parts with an unavoidable intersection property: for any finite coloring of , some color class intersects both parts infinitely many times, after an appropriate shift (translation). We establish a polynomial extension of this result, proving that such intersections persist under polynomial shifts in any dimension. Let be non-constant polynomials with positive leading coefficients and for every . We construct a partition of into an arbitrarily fixed finite number of pieces such that for any coloring of with finitely many colors, there exist and a single color class that meets all partition pieces after shifts by in each of the coordinate directions, for every and infinitely many values . Our proof exploits Weyl's equidistribution theory, Pontryagin duality, and the structure of polynomial relation lattices. We also prove some finite analogues of the above results for abelian groups and .

Paper Structure

This paper contains 12 sections, 17 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof ideas and techniques
  3. Higher dimensions -- Proof of Theorem \ref{['HD']}
  4. Partition and an induced coloring
  5. Preliminary lemmas
  6. Proof of Theorem \ref{['HD']}
  7. Polynomial shifts -- Proof of Theorem \ref{['thmpolynomial-main']}
  8. Proof of Theorem \ref{["lm2.2'"]}
  9. Proofs of Lemmas \ref{['lem:H=Hprime']} and \ref{['prop:weyl-H']}
  10. Finite abelian groups -- Proof of Theorem \ref{['thm3.1']}
  11. Special linear groups -- Proof of Theorem \ref{['thm3.2']}
  12. Acknowledgements

Key Result

Theorem 1.1

There exists a subset $E\subseteq \mathbb{N}$ with the following property. For every finite coloring of $\mathbb{N}$ with $t$ colors, $\mathbb{N}=\bigcup_{j=1}^t F_j$, $t\in\mathbb{N},$ there exist a single color class $F_m$, $1\le m\le t$, and $k\in \mathbb{N}$ such that both $(F_m+k)\cap E$ and $(

Figures (1)

  • Figure 1: The partition structure of $(0, 1)$ into bands $S_j$ and tiles $T_{j,i}$

Theorems & Definitions (29)

  • Theorem 1.1: Raimi Raimi
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 19 more