在丢番图逼近的研究中，连分数展式和整数进制展式是实数的表示理论中非常重要的工具。这两种表示方式所具有的度量性质和分形性质具有很强的理论意义和广泛的应用价值。本项目拟利用维数理论、数论研究的特点和技巧研究连分数展式和整数进制展式在丢番图逼近与正规性中的相互关联性： (1) 研究实数的连分数表示与整数进制表示逼近实数的效率，进而研究一般的区间映射引导的实数表示方式对实数的逼近效率。(2) 研究三分Cantor集上实数的连分数表示的基本性质。(3) 研究同一实数在连分数表示与整数进制表示下的正规性。. 本研究致力于探索实数在一种表示方式具有特定性质的情况下另一种表示方式的度量性质和分形性质，进一步揭示实数不同表示方式间的深层次联系，利用研究这些问题所发现的方法和技巧来推动分形几何、 丢番图逼近与数的表示理论领域间的交叉发展。

In the study of Diophantine approximation, the continued fraction expansion and integer base expansion are important tools in the representation theory of real number. The measurement and fractal properties of the two representations have a strong theoretical significance and extensive application value. This project intends to research mutual relevance of the continued fraction expansion and the integer base expansions in the Diophantine approximation and normality. Firstly, we will research the efficiency of approximating real numbers by continued fraction and integer base expansion as well as the efficiency of approximating real numbers by the two representations of real number induced by two general different interval maps. Secondly, we will research the basic properties of continued fraction expansion for the real numbers in middle Cantor set. Thirdly, we will research the normality between the continued fraction expansion and the integer base expansion of real numbers..This project is aimed at exploring the measurement and fractal properties of one representation of real numbers whose another representation have specific properties, and revealing the deep relations between the different representations of real numbers. Meanwhile, it is a long hope that the new methods and techniques established in the study of these questions can promote the cross development between fractal geometry, the representations theory of real numbers and the fields of Diophantine approximation.