Diophantine逼近是数论中的重要研究分支，重点研究实数被有理数逼近的程度，而Diophantine逼近指数作为实数上的函数是刻画这种逼近程度的重要的指标。近几十年来，随着分形几何的迅速发展，分形集上的Diophantine逼近逐渐成为数论研究的焦点问题，同时，分形维数和测度理论为解决Diophantine逼近相关问题提供有力的研究方法，思想和工具，具有重要的理论意义。.本项目拟用分形维数和测度作为工具，重点探讨一下几个问题：1. 分形集上Diophantine逼近指数及相应水平集的分形测度和维数；2. 随机扰动的自相似集上μ-可很好逼近集的分形维数；3. 分形集上一类可很好逼近集的Duffin-Scheaffer型“0-1”律。上述问题的解决有助于推动分形几何和Diophantine逼近双方学科领域的交叉发展。

Diophantine approximation is an important branch in the study of number theory, and mainly focuses on what extent can real numbers be approximated by rational numbers. The exponents of Diophantine approximation as a function defined on the set of real numbers is an important quantitative indicators of such approximation. Over the past decades, with the rapid development of fractal geometry, the Diophantine approximation in fractal sets gradually become a focus problem in the study of number theory. Meanwhile, fractal dimensions and measures theory provide a powerful method, idea and tool to solve the related problems of Diophantine approximation, which is of great theoretical significance. .By using fractal dimension and measure as the main tool, we study the following subjects in this project: 1. the study of the exponents of Diophantine approximation on fractal sets as well as fractal measure and dimension of level sets related to exponents ; 2. the study of fractal dimension of the intersection of μ-well approximable set with the rotated self-similar sets; 3. the study of the analogue of Hausdorff measure version of the Duffin-Scheafer "0-1" law for a class of well approximable sets on fractal sets. The solution of the problems above can help to promote the interdisciplinary development of fractal geometry and Diophantine approximation.