谱与Tilings关系中的测度估计和平移对

本文将在两种特有的情形下研究谱与tilings之间的关系.首先,估计和比较谱与tilings关系中集合的Lebesgue测度,这包括一些不能直接用密度方法所得结果的推广,以及在正交对、填充对与覆盖对中集合的Lebesgue测度的比较.其次,明确了平移对(D,∧+Г)与(D+Г,∧)之间的一些谱与tilings关系.这里的研究是基于谱与tilings的基本性质,与共轭Fuglede猜想密切相关.

Measure Estimates and Translative Pairs in the Spectra-Tilings Relations

In the present paper we will study the relations between spectra and tilings in two special cases.We first estimate and compare the Lebesgue measure of sets in the spectra-tilings relations.This includes a generalization of several results when the density method fails to apply,and the comparison of Lebesgue measure of sets among the orthogonal pairs,packing pairs and covering pairs.We then clarify some spectratilings relations between the translative pairs(D,A +Γ) and(D +Γ,A).The research here is based on the fundamental properties of spectra and tilings,and is closely related to the dual Fuglede conjecture.