  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.
