平面上测度的谱性与tile

The project mainly study the spectrality of fractal measures and the tile property of self-affine sets in the plane. We intend to initiate a study on the structure of spectra and tiles by developing the existing tools and techniques and borrowing the knowledge from group theory, number theory and combinatorics. The main contents are as follows: (1) The spectrality of fractal measures in the plane, especially the Moran measures. Undering the assumption of Hadamard triple, we will investigate the relationship between the overlaps of Moran sets and the spectrality of Moran measures. This is an effective way to find spectral Moran measures; (2) The eigen-matrix prolem of spectral measures in the plane. We intend to explore the eigen-matrix of spectra for some spectral measures with the help of graph direct system. This will build the foundation of the furthure research on the Beurling dimension theory of spectra; (3) The structure of self-affine tile digit sets. We want to define a class of functionals which have some properties similar with cyclomonic polynomials. Then we can analyze the property of the mask polynomials and classify the sufficient and necessary condition for a digit set to be a self-affine tile digit.

本项目主要研究平面上分形测度的谱性质以及自仿集的tile性质。 我们将在发展现有工具和技巧的基础上，引入群论，数论，组合数学等方面的知识研究谱和tile的结构。具体研究内容包括：（1）平面上分形测度的谱性，主要研究对象是Moran 测度。我们拟从Hadamard 三元组出发，探究Moran集重叠程度与Moran测度谱性的关系，确定Moran 谱测度；（2）平面上谱测度的特征矩阵问题。我们将引入图递归系统分析刻画测度谱的特征矩阵，这也将为深入研究谱的Beurling维数理论奠定基础；（3）平面上自仿tile数字集的结构。我们拟定义平面上的一类泛函， 使其与分圆多项式有某些相似的特性。通过分析mask多项式的属性，刻画数字集成为自仿tile数字集的充要条件。

本项目主要研究Moran测度的谱性质以及自仿集的tile性质。我们将在发展现有工具和技巧的基础上，引入调和分析，群论，组合数学等方面的知识研究谱和tile的结构。具体研究内容包括：（1）一维分形测度的谱性，主要研究对象是具有三个数字集的Moran测度。我们拟从Hadamard三元组出发，探究支撑集的收敛速度与Moran测度谱性的关系, 这是构造谱测度的有效方法（2）讨论一维上支撑在[0,1]上的Moran测度的谱性，若测度满足equiv-positive性质, 则Hadamard三元组可以保证其谱性. 对于不满足equiv-positive性质的Moran测度，我们证明了其谱性与Bourgain的sine求和猜想直接相关；（3）利用一维投影的思想，刻画了对应对角矩阵pI的自相似tile数字集。