在实数中，对有理数稠密性的量化性质研究，使得丢番图逼近成为数论研究的核心内容之一；在动力系统中，对轨道稠密性的量化性质研究是揭示系统本质属性的重要手段。本项目拟利用分形的方法研究经典丢番图逼近和动力系统中轨道分布的度量理论：1) 在经典逼近中，围绕数论中未解决的著名问题，包括Littlewood猜测、Duffin-Schaeffer猜测，首先开展其弱化形式的研究，如有约束条件的Littlewood猜测、加强的发散性条件下的Duffin猜测；2) 在动力系统中，研究轨道的丢番图属性，包括轨道返回起始点、轨道逼近给定点列、给定轨道的分布等。 本研究致力于探索解决经典丢番图逼近中问题的途径和方法；建立动力系统中丢番图逼近的度量理论。由于经典逼近与动力系统中轨道分布的密切关联，希望能借助动力系统中建立的理论解决经典逼近中的问题；并利用在研究丢番图逼近问题中发现的新思路来推动分形理论的发展。

In the real number field, the quantitative study on the density property of rational numbers makes Diophantine approximation a core field in number theory; in dynamical systems, the quantitative study on the density property of the orbits is an important tool to analyze the essential features of the corresponding dynamical system. In this project, we apply the methods developed in Fractal Geometry to study the metric properties of the classic Diophantine approximation and the distribution of the orbits in dynamical systems: 1). In the classic approximation, we focus on some famous open conjectures, including Littlewood conjecture and Duffin-Schaeffer conjecture, to study the weaken form of these conjectures, for instances, Littlewood conjecture with restrictions, Duffin-Schaeffer conjecture with stronger divergence conditions; 2). In dynamical systems, we study the Diophantine properties of the orbits, including the quantitative properties of the orbits returning back to the initial point, the properties of the orbits when approximating some given sequence as well as the distributions of the orbits of a given point. This project is aimed at exploring the possible ways to solve the problems in the classic Diophantine approximations; setting up metric theory for Diophantine approximation in dynamical systems. Due to the intimate relations between classic approximation and the distribution of the oribts in some dynamical systems, we also attempt to find solutions to the questions in classic Diophantine approximation from the point of view of dynamical systems. Meanwhile, it is a long hope that the new methods established in studying these Diophantine questions can promote the development of Fractal Geometry.