The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work.

在Diophantine逼近中，由球定义或由矩形定义的limsup集出现在最基本的层次上：一个遵循Dirichlet定理，另一个遵循Minkowski定理。前者的量度理论已被深入研究，而后者的理论则相当不完整，即使是经典的丢番图近似也是如此。本文旨在为矩形定义的limsup集的Hausdorff理论建立一般原理。通过引入一种称为矩形的普遍存在的表示法，提出了从矩形到矩形的传质原理，即，如果由一系列大矩形定义的limsup集具有完整的量度属性，则可以将此完整量度属性转换为limsup的完整的Hausdorff量度/维数属性通过将这些大矩形缩小为较小的矩形来定义的集合。在此，对于较大的limsup集，其全量度属性在假定或不假定矩形无处不在的情况下进行。与Beresnevich和Velani的地标性工作（Ann Math（2）164（3）：971–992，2006）一起建立了球到球的转移原理，建立了用于度量Diophantine近似的连贯的Hausdorff理论。除了完成同时丢番图近似，线性形式的尺寸理论外，