Tiling by translates of a function: results and open problems

TL;DR

This paper surveys and advances the understanding of how functions can tile the real line through translations, exploring various densities, distributions, and periodicities, while highlighting open problems in the field.

Contribution

It provides new results on the structure of tilings by translates of functions, including cases of bounded and unbounded density, and discusses open problems.

Findings

Characterization of tilings with bounded density

Results on non-periodic and zero-level tilings

Open problems in tiling structures

Abstract

We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $\Lambda \subset \mathbb{R}$, if we have $\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of $\mathbb{R}$ by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.