近几十年来，随着实数非整数基表示研究的深入，其与分形几何、动力系统、遍历理论、图论等学科的联系一再被揭示，逐渐成为国内外同行关注的热点。综合分形几何、动力系统、遍历理论等方法和技巧，项目拟首先研究实数非整数基表示中的三类集合：1. 由满足某条件的基组成的集合；2. 由点组成的集合，该集合中每个点有相同的固定个数的q-展式；3. 由点的所有q-展式组成的集合。我们研究前两类集合的代数刻画、分形性质、拓扑结构；增长率、分形维数、不可数性和普适展式是第三类集合的复杂性度量，我们研究这些复杂性度量之间的关系。进而，拟把对这三类集合的研究推广到分形几何中不满足开集条件的自相似集中的类似三类集合上。这些问题体现了分形几何、实数非整数基表示、动力系统等学科的交叉，需要融合不同学科的思想方法和发展新技巧。研究成果对进一步丰富和推进分形几何及实数非整数基表示的发展具有十分重要的意义。

Over the past few decades, representations for real numbers in non-integer bases has been extensively studied and its study revealed a number of connections with fractal geometry, dynamics, ergodic theory and graph theory, etc. It has received a lot of attention. With the methods and techniques of fractal geometry, dynamics and ergodic theory, we will first investigate three classes of sets, which are: 1. a set composed of some bases satisfying some conditions; 2. a set composed of some points having fix number q-expansions; 3. a set composed of all q-expansions of a point. We study the equivalent characterization, fractal properties and topological structure of the first two sets. The growth rate, fractal dimension, uncountability and universal expansion reveal the complexity of the last set. We will study the relation among them. Furthermore, we will investigate the similar three classes of sets of self-similar sets with overlaps in fractal geometry. These research questions reveal the overlaps of fractal geometry, representations for real numbers in non-integer bases and dynamics, to solve them we need to integrate the methods of different subjects and develop new techniques. The investigation of this project is of important significance for advancing fractal geometry and representations for real numbers in non-integer bases.