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

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.

To appear in International Mathematics Research Notices (IMRN)

Cited by in corpus (7)

• Tiling by translates of a function: results and open problems
• The structure of translational tilings in ${\mathbb{Z}}^d$
• Gabor orthonormal bases, tiling and periodicity
• Spectral sets and weak tiling
• An example concerning Fourier analytic criteria for translational tiling
• Functions tiling simultaneously with two arithmetic progressions
• Spectrum is rational in dimension one