On characterizations of spectra and tilings

In a recent paper, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove a conjecture of Jorgensen and Pedersen by Kellerʹs criterion. Different techniques to prove these facts have also been developed by Kolountzakis, Iosevich and Pedersen. The primary aim of this paper is to present an elementary method of describing certain characterizations of spectra and tilings. To illustrate this method, we first give a simple proof of this characterization. We then use the method to derive some characteristic results connected with the dual Fugledeʹs spectral-set conjecture. The results here extend several known conclusions in a simple manner.