An example concerning Fourier analytic criteria for translational tiling

Abstract

It is well known that the functions $f\in L^1({\mathbb{R}}^d)$ whose translates along a lattice $\Lambda$ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a discrete set $\Lambda \subset \mathbb{R}$ (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions $f,g\in L^1(\mathbb{R})$ whose Fourier transforms have the same set of zeros, but such that $f+\Lambda$ is a tiling while $g+\Lambda$ is not.