| Spectral pairs have certain relations to Tiling pairs. Both have a direct appli-cation in the theory of wavelets and in the theory of trigonometric approximation. The relations between spectral set and tile, as well as spectrum and Tiling set, are rather mysterious. There are several conjectures to clarify their relations. In the dual Fuglede conjecture, it is known that the existent sets Ω and D must satisfy m(Q)m(D)=1. This is a necessary condition for the dual Fuglede conjecture. In the research of relations between spectra and Tilings, the Lebesgue measure of sets inside them must satisfy certain relations which we need to know at first. In this re-gard, the density method plays an important role, but it only provides us the partial results on the Lebesgue measure of sets inside spectra and Tilings when the density of the corresponding discrete set exists. The detailed estimation and comparison of sets in the spectra-Tilings relations are not completely determined.In this article, we shall discuss these questions in two part:Part1: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 andcovering pairs. We then clarify some spectra-Tilings relations between the translative pairs (D, A+Г) and (D+Г, A).Part2:We shall give an application of the density method in spectra-Tilings relations. The research here is based on the fundamental properties of spectra and Tilings, By applying the density method, we estimate the Lebesgue measure of sets in the spectra-Tilings relations. This is necessary for the further investigate on the relations between spectra and Tilings.The research here is based on the fundamental properties of spectra and Tilings, and is closely related to the dual Fuglede conjecture. |
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