Tiling by translates of a function: results and open problems

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.

References in corpus (2)

• A Simple Crystalline Measure
• An example concerning Fourier analytic criteria for translational tiling

Cited by in corpus (5)

• The Fuglede conjecture for convex domains is true in all dimensions
• The structure of translational tilings in ${\mathbb{Z}}^d$
• Gabor orthonormal bases, tiling and periodicity
• An example concerning Fourier analytic criteria for translational tiling
• Functions tiling simultaneously with two arithmetic progressions