丢番图逼近是数论的一个重要分支，它主要量化地研究有理数的分布性质，即无理数能够被有理数逼近的程度。在度量丢番图逼近的研究中，分形测度和分形维数起着十分重要的作用，是目前国内外研究的热点问题，具有重要的理论意义。. 本项目研究有限制的丢番图逼近问题，即用具有一定约束条件的有理数去逼近无理数。我们将研究有限制的丢番图逼近的质量转移原理；研究有限制丢番图逼近的Khintchine和Jarník定理；研究丢番图逼近中下限集的尺度（测度及维数）以及研究三分Cantor集上的丢番图逼近问题。我们致力于探索有一定约束条件的有理数的分布情况，建立限制丢番图逼近以及下限集的度量理论，结合经典的质量转移原理研究相应的分形性质。

Diophantine approximation is an important branch of number Theory, which aims at quantifying the property of the distribution of rational numbers, that is the approximation of irrational numbers by rational ones. Fractal measures and fractal dimensions play a very important role in the study of metric Diophantine approximation, which is hot topic of domestic and foreign research and have an important theoretical significance.. This project study the problem of restricted Diophantine approximation, that is the approximation of irrational numbers by rational numbers with satifying some restricted conditions. We will study Mass Transference Principle for restricted Diophantine approximation, study restricted Diophantine approximation of Khintchine and Jarník theory, study the level (measure and dimension) of the liminf set of Diophantine approximation and study the problem of Diophantine approximation on Cantor set. We are committed to exploring the distribution of certain constraints rational numbers, to establish metric theory between restrictioned Diophantine approximation and the liminf set, combining with the classical Mass Transference Principle to study corresponding properties of fractal.