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.

Locations

• Revista Matemática Iberoamericana - View - PDF
• arXiv (Cornell University) - View - PDF