Cantor集的平移交与beta展式的研究

Intersections of Cantor set with its translations and beta expansions got widespread interest in mathematical branches, such as fractal geometry, dynamical systems, combinatorics and so on. Recently people found that the intersection of Cantor set with its translation can be a self-similar set, they also found close connection between the intersections of Cantor sets and the unique beta expansions. The methods and techniques from beta expansions are used in the intersections of Cantor sets, and the other way around, the methods and techniques from fractal geometry are used to deal with problems in beta expansions. We will combine these two methods to study the algebraic and geometric properties of the set of points which have exactly m beta expansions. Moreover, we will analyse in a fractal way that the set of points which have a unique beta expansion, and calculate its Hausdorff dimension. Solving related problems will make great progress in the study of intersections of Cantor sets and beta expansions.

Cantor集的平移交与beta 展式的研究在分形几何、动力系统、组合数学等数学分支中引起广泛的兴趣。近年来人们惊奇地发现Cantor集的平移交有可能是一个自相似集，并且在研究过程中发现其与唯一beta展式之间有着密切联系。处理经典beta展式的方法与技巧被借鉴到Cantor集的平移交上，与此同时，处理分形几何的技巧与手段也同样被用来处理beta展式中的相关问题。我们将结合这两者讨论有且仅有m个beta展式集合的代数与几何特征，并讨论与之相应的Cantor集平移交的自相似性。另外我们将对唯一beta展式集合进行分形分析，并计算其Hausdorff维数。解决与之相应的问题对研究Cantor集的平移交与beta展式有着重要意义。

Cantor集的平移交与$\beta$ 展式的研究是分形几何、动力系统等领域中的重要研究课题，其研究涉及到数学的很多分支，如概率论、遍历理论、组合数学、符号动力系统、拓扑学、数论等。本项目主要研究了Cantor集平移交的自相似性和唯一$\beta$展式集合的Hausdorff维数。.. 在Cantor集平移交的自相似性方面，我们给出了一类广义Cantor集为齐次生成自相似集的充分必要条件。该结果可以用来讨论两个广义Cantor集平移交的自相似性。 在唯一$\beta$展式方面，我们给出了对几乎所有的$\beta>1$ 集合$U_{\beta,N}$的Hausdorff维数公式，其中$U_{\beta,N}$为所有具有唯一$\beta$展式的数的全体。.. 本项目基本完成计划目标。关于平移量不具有唯一表示时，Cantor集平移交的自相似性仍有待进一步研究。