On a criterion of universal spectra

Abstract

For positive integers N1,N2, …, Nn, let L = ℤn + A and Λ = N1ℤ × … × Nnℤ + gT be two rational periodic sets, where A ⊂ (1/N1)ℤ × … × (1/Nn)ℤ and gT ⊂ ℤn are finite sets with (A−A)∩ℤn = {0} and (gT−gT)∩(N1ℤ×…×Nnℤ) = {0}. In this note, we shall determine conditions under which the tiling set L has universal spectrum Л. We first obtain a criterion of universal spectra. This criterion combined with the properties of compatible pair yields many necessary and sufficient conditions for Л to be a universal spectrum for L. We then show that, under some mild additional conditions, the conjecture of Lagarias and Szabo is true. The results here extend the corresponding results of Lagarias, Szabo and Wang.