作为现代数学中一个新的分支，分形几何与与动力系统，遍历理论以及Diophantine逼近都有着密切的联系，于是分形几何与其他学科的交叉研究是现代分形理论研究的一个重要增长点。Diophantine逼近是数论研究的一个重要分支，目前关于Diophantine逼近研究的注意力主要集中在对逼近误差进行估计和度量，因此由各种误差函数所确定的不可很好逼近集就成为人们研究的重点。本项目拟利用分形集的维数理论来度量Diophantine逼近中满足逼近条件限制的不可很好逼近集的大小，具体包括：（1）研究齐次Diophantine逼近中一类不可很好逼近集的Hausdorff维数；（2）研究与Littlewood猜想相关的一类例外集的Hausdorff维数；（3）研究非齐次Diophantine逼近中一类不可很好逼近集的Hausdorff维数。上述问题的解决将有利于推动双方学科的交叉发展。

As a new branch of modern mathematics, Fractal Geometry is closely associated with Dynamical Systems, Ergodic Theory and Diophantine Approximation, so the cross study of Fractal Geometry with other disciplines is an important growth point of modern fractal theory. Diophantine Approximation is an important branch of Number Theory. The current research of Diophantine approximation mainly focuses on the estimation and measure of approximation error, therefore the badly approximable sets determined by a variety of error functions become the main issues of research. This project intends to use the dimension theory of fractals to measure the size of the badly approximable sets which satisfy some approximation conditions, specifically including the following three problems: (1) Explore a class of badly approximable sets in homogeneous Diophantine approximation, and estimate the Hausdorff dimensions of these sets; (2) Study Hausdorff dimensions of a class of exceptional sets related to the Littlewood cejecture; (3) Explore a class of badly approximable sets in inhomogeneous Diophantine approximation, and estimate their Hausdorff dimensions. The solution of the problems above will be conducive to promoting the cross development of both disciplines.