On non-periodic tilings of the real line by a function

TL;DR

This paper investigates the conditions under which functions can tile the real line through translations, revealing that unbounded support allows non-periodic tilings, but finite local complexity enforces periodicity.

Contribution

It proves that unbounded support enables non-periodic tilings, while finite local complexity still guarantees periodicity, extending understanding of tiling structures.

Findings

Unbounded support allows non-periodic tilings.

Finite local complexity implies periodicity.

Positive, compactly supported functions only tile periodically.

Abstract

It is known that a positive, compactly supported function $f \in L^1(\mathbb R)$ can tile by translations only if the translation set is a finite union of periodic sets. We prove that this is not the case if $f$ is allowed to have unbounded support. On the other hand we also show that if the translation set has finite local complexity, then it must be periodic, even if the support of $f$ is unbounded.