An example concerning Fourier analytic criteria for translational tiling

TL;DR

This paper constructs a specific example of a discrete set where Fourier analytic criteria fail to characterize translational tiling, challenging existing theorems that rely on zero sets of Fourier transforms.

Contribution

It provides a counterexample demonstrating that zero set conditions are insufficient for characterizing tiling for certain discrete sets.

Findings

Counterexample set $\\Lambda$ where Fourier zero set criteria fail

Existence of functions with identical Fourier zeros but different tiling properties

Challenges to Fourier-based tiling characterization methods

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.