Functional tilings and the Coven-Meyerowitz tiling conditions

TL;DR

This paper explores a relaxed version of the Coven-Meyerowitz tiling conjecture involving nonnegative functions, revealing that the original tiling conditions do not always apply in this broader context, which could impact future tiling counterexamples.

Contribution

It introduces a functional tiling framework that generalizes the Coven-Meyerowitz conditions and demonstrates their limitations within this broader setting.

Findings

Coven-Meyerowitz conditions do not always hold for functional tilings

Functional tilings can serve as potential counterexamples to the conjecture

The study broadens understanding of tiling conditions beyond sets

Abstract

Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group $\mathbb{Z}_M$. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions $f,g$, such that $f(0)=g(0)=1$, $f\ast g=\mathbf{1}_{\mathbb{Z}_M}$ is a functional tiling, and $f, g$ satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.