AI 中文总结
本文给出Ahlfors--David正则度量测度空间存在指定测度且直径最优划分的充分条件，通过二进立方体树的算法填充构造证明，并发展技术构造此类空间。
AI 中文摘要
本文在Ahlfors--David正则度量测度空间上找到了一个充分条件，使得该空间能够被划分为具有指定测度且直径最优（至多相差一个常数）的部分。证明使用了二进立方体的构造。该过程是算法化的：通过二进立方体树上的填充过程，逐块切割出各个部分。我们引入了具有连通二进立方体分解的空间的概念，并证明了这类空间具有上述类型的划分。随后，我们发展了若干技术来构造此类空间，并展示了这类空间的一些自然例子。
英文摘要
In this paper we find a sufficient condition on an Ahlfors--David regular metric measure space under which it admits a partition into parts of prescribed measures and optimal (up to a constant) diameters. The proof uses the construction of dyadic cubes. The process is algorithmic: the pieces are cut out one by one via a filling procedure on the tree of dyadic cubes. We introduce the notion of spaces which admit a connected dyadic cube decomposition and prove that they admit a partition of the kind described above. We then develop several techniques to obtain such spaces and show some natural examples of this type.
Comments23 pages, 4 figures
