测度谱的平铺性质
Tiling properties of spectra of measures
摘要：
我们调查平铺性质的频谱措施，即，在$\br$中设置$Λ$，使得$\{e^{2πi λx}: λ\inΛ\}$形成$L^2(μ)$中的正交基，其中$μ$是$\br$上的某个有限Borel测度。这类测度包括有界Borel子集上的勒贝格测度、有限原子测度和一些分形Hausdorff测度。我们表明，各种类别的这种频谱的措施有平移平铺性质。这就得到了分形测度谱的一些特殊性质，证明了具有Coven-Meyerowitz性质的有限集的可补集和谱的存在性，以及在Hadamard对的大小为2，3，4或5的情况下可补Hadamard对的存在性.在Fuglede猜想的背景下，我们证明了任何谱集是一个瓦片，如果谱的周期是2，3，4或5。
英文摘要：
We investigate tiling properties of spectra of measures, i.e., sets $Λ$ in $\br$ such that $\{e^{2πi λx}: λ\inΛ\}$ forms an orthogonal basis in $L^2(μ)$, where $μ$ is some finite Borel measure on $\br$. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard pairs in the case of Hadamard pairs of size 2,3,4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2,3,4 or 5.
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