An example concerning Fourier analytic criteria for translational tiling

• Nir Lev ^[1]
  1. [1] Bar-Ilan University

     Bar-Ilan University

     Israel

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 6, 2022, págs. 1975-1991
• Idioma: inglés
• DOI: 10.4171/RMI/1318
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• Resumen

  • It is well known that the functions f∈L1(Rd) whose translates along a lattice ΛΛ 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 +Λ is a tiling while g+Λ is not