On Characterizations Of Spectra And Tilings In Square Integrable Function Space

Spectra and tilings play an important role in analysis and geometry respectively, the relationship between them has led to a lot of conjecture, for example Fuglede spectral set conjecture, Reeds and Wang established a characterization of spectra and tilings that can be used to prove a conjecture of Jorgensen and Pedersen by Keller’s criterion. Different techniques to prove these facts have also been respectively developed by Iosevich, Pedersen and Kolountzakis later. For a common goal, the preliminary studies are trying to figure out the relationship between spectra and tilings. On the basis of former achievements, this paper continues to study spectra and tilings and presents an simple method of describing certain characterizations of spectra and tilings. The results here extend several known conclusions in a simple manner. Finally, this paper also shows certain characterizations of period sets.