序列的分布及相关的测度与分形性质

The classical Diophantine approximation deals with the approximation of irrational numbers by rational numbers, the most important problem in this field is to know how well an irrational number can be approximated by rational numbers, this question amounts to the study of distribution of the sequence of rational numbers. In general, the study of the distribution of certain sequence, together with the related measures and fractals, constitute the key of the metric theory of Diophantine approximation. In this project, we will embark on a probing in depth of the distribution of the orbit determined by a fixed point in certain dynamical system and the measures and fractals related to it. It involves the investigations of: 1) the size of the classical set of well approximable numbers in the study of irrational rotation, the relation between the Lebesgue measure as well as Hausdorff dimension of such set and irrational exponent, the adaptability of the mass transference principle; 2) the characterization of whether an orbit of dyadic dynamical system, as well as some Markov dynamical systems with an infinite number of inverse branches, to be a Borel-Cantelli sequence. This project is closely related to the theory of classical Diophantine approximation on the one hand, and provides quantitative results for certain properties of the distribution of the orbits in dynamical systems on the other. In particular, the study of the distribution of the orbit determined by an irrational number will underpin and extend the metric theory of inhomogeneous Diophantine approximation.

经典丢番图逼近指用有理数逼近无理数，其首要问题是研究无理数能够被有理数逼近的程度，本质是研究有理数序列的分布性质。作为经典丢番图逼近理论的拓展，给定序列的分布及相关的测度与分形性质构成了度量丢番图逼近的核心研究内容。本项目拟研究对给定点在动力系统作用下的轨道序列的分布性质及相关的测度与分形结构。具体内容包括：1）无理旋转中，可很好逼近点集的尺度问题，该集合的Lebesgue测度、Hausdorff维数与无理指数之间的关系，以及质量转移原理的有效性等；2）加倍系统和部分具有无穷Markov分划的动力系统中，轨道是否为Borel-Cantelli序列的刻画。此研究一方面与经典丢番图逼近理论有密切的关联，另一方面是对动力系统中轨道分布的定性理论的定量刻画。特别地，对无理旋转轨道分布的研究，其目的为深入和发展非齐次丢番图逼近的度量理论。

近年来，动力系统理论在研究丢番图逼近方面的重大突破, 使得动力系统中的丢番图逼近理论研究成为国际上度量丢番图逼近理论发展的重要方向。对无理旋转，基于Legendre定理，进一步完善了在Ostrowski展式下的Borel-Cantelli引理。在度量丢番图逼研究中，基于Khintchine–Jarník–Besicovit定理，改进了若干Dirichlet non-improvable及well-approximable等渐进丢番图逼近问题，所得结果进一步完善了已有结果，更具一般性，并且给出了若干Cantor子集构造技巧。为获得适用于无穷符号空间Luroth系统中的轨道性质，获得了Alhaph-Luroth展式和Galmbos提出的关于快速增长集的维数结果，所得结果改进了已有结果。对多变量连分数的二维Gauss-Kuzmin问题，得到了误差项分布函数的收敛率，更加清晰地理解了Gauss-Kuzmin问题的本质。