丢番图逼近与数的表示理论中相关分形集交集的研究

我们常对关于实数的两类性质很感兴趣：一类是在经典的丢番图逼近中考虑实数可以被有理数很好地逼近的性质；另一类则是在数的表示理论中出现的一些性质，例如在数的某个展式中，给定某个字符出现的数字频率或考虑该展式下前n个字符中连续出现某个相同字符的最大的个数等。分别研究具备以上两类性质的实数集的大小，一直以来都是度量数论、分形几何以及动力系统等学科所考虑的重要课题之一。. 本项目旨在将以上两种实数分类的方式相结合，拟研究同时满足以上两类性质的实数集的大小，从而给出实数的一种更为细化的分类。具体而言，本项目将从以下三个方面展开研究：1.研究连分数系统下Jarník集与Erdös-Rényi集交集的分形结构；2.考察Jarník集与Besicovitch集交集的分形维数大小；3.考虑在连分数系统中数字频率限制下相关集合的大交性质。

We are interested in two properties of real numbers: the first one is the property of being well-approximated by rational numbers, which is arisen in the classical Diophantine approximation; the second property appears in the representation theory of reals, concerning the numbers having given digit frequencies in some expansions or the longest run of some same symbol in the first n digits of the corresponding expansions. The study for the sizes of the sets of reals satisfying the above two kinds of properties respectively have always been one of the important tasks in the subjects such as metric number theory, fractal geometry and dynamical system and so on.. This project intends to combine these two ways of classifying the real numbers, and investigates the sizes of the sets of reals satisfying the above two properties simultaneously, so that we can provide a finer classification of real numbers. More specifically, we will study the following three aspects: 1. Study the fractal structures for the intersections of Jarník sets and Erdös-Rényi sets in continued fraction systems; 2. Explore the fractal dimensions for the Jarník sets and Besicovitch sets; 3. Consider the large intersections properties under digit frequencies constraints for some related sets in continued fraction expansions.