For positive integers N1,N2,...,Nn, let L = Z^n ＋ A and A =- N1Z x -…… x NnZ ＋ F be two rational periodic sets, where A ∩ （1/N1）Z x ... x （1/Nn）Z and F ∩ Z^n are finite sets with （A- A）∩Z^n = {0} and （F...For positive integers N1,N2,...,Nn, let L = Z^n ＋ A and A =- N1Z x -…… x NnZ ＋ F be two rational periodic sets, where A ∩ （1/N1）Z x ... x （1/Nn）Z and F ∩ Z^n are finite sets with （A- A）∩Z^n = {0} and （F- F）N （N1Z x.. x NnZ） = {0}. In this note, we shall determine conditions under which the tiling set L has universal spectrum A. We first obtain a criterion of universal spectra. This criterion combined with the properties of compatible pair yields many necessary and sufficient conditions for A to be a universal spectrum for L. We then show that, under some mild additional conditions, the conjecture of Lagarias and Szab5 is true. The results here extend the corresponding results of Lagarias, Szabo and Wang.展开更多
基金Supported by National Natural Science Foundation of China (Grant No. 11171201)the Fundamental Research Fund for the Central University (Grant No. GK201001002)
文摘For positive integers N1,N2,...,Nn, let L = Z^n ＋ A and A =- N1Z x -…… x NnZ ＋ F be two rational periodic sets, where A ∩ （1/N1）Z x ... x （1/Nn）Z and F ∩ Z^n are finite sets with （A- A）∩Z^n = {0} and （F- F）N （N1Z x.. x NnZ） = {0}. In this note, we shall determine conditions under which the tiling set L has universal spectrum A. We first obtain a criterion of universal spectra. This criterion combined with the properties of compatible pair yields many necessary and sufficient conditions for A to be a universal spectrum for L. We then show that, under some mild additional conditions, the conjecture of Lagarias and Szab5 is true. The results here extend the corresponding results of Lagarias, Szabo and Wang.
