An example concerning Fourier analytic criteria for translational tiling

It is well known that the functions f ∊ L¹(R^d) whose translates along a lattice Λ 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 Λ ⊂ R (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions f; g ∊ L¹(R) whose Fourier transforms have the same set of zeros, but such that f + Λ is a tiling while g + Λ is not.