丢番图逼近研究实数被有理数逼近的精度问题，实质就是研究有理数的分布性质。Dirichlet定理是关于有理数分布性质的最根本的结果，是度量丢番图逼近理论的起点。关于Dirichlet定理的可改进性的研究是目前数论、分形理论等领域专家关注的热点问题之一。.本项目拟展开对Dirichlet定理及其延伸的可改进性理论的研究。具体的，我们将研究Dirichlet可改进集的分形性质；研究连分数中连续部分商乘积的渐进行为；研究非齐次的Dirichlet定理的可改进性问题。本项目旨在发展一致丢番图逼近理论以及完善连分数中部分商乘积的维数理论。

Diophantine approximation studies the accuracy of a real number approximated by rationals, which is in essence to study the distribution properties of rational numbers. Dirichlet's theorem is the most fundamental result about the distribution properties of rationals and the starting point in metric theory of Diophantine approximation. The study on the improvability of Dirichlet's theorem is one of the hot issues in the field of number theory, fractal theory, etc..This project intends to initiate a study on the improvable theory of Dirichlet's theorem and its extension. More precisely, we will study the fractal properties of Dirichlet improvable sets, study the asymptotic behavior of the product of consecutive partial quotients in continued fractions，study the improvability of inhomogeneous Dirichlet's theorem. This project aims at developing uniformly Diophantine approximation theory and improving the dimension theory of the product of partial quotients in continued fractions.