AI 中文总结
该研究证明，对有限稳定的分形维数定义，度量空间紧子集内存在与自身维数相同的收敛点序列，部分情况下该序列可同时对应多种维数定义。
AI 中文摘要
对于有限稳定的各类分形维数定义，包括上盒维数、上中间维数和Assouad谱，我们证明：给定度量空间（通常为Rⁿ）的紧子集E，存在E中收敛的点序列，其维数与E本身的维数相同。此外，在某些条件下，一个序列可同时对应多种维数定义的E的维数，例如自仿射集E中存在单个收敛序列，其Assouad谱或中间维数值与E本身相同。
英文摘要
For various definitions of fractal dimension that are finitely stable, including upper box dimension, upper intermediate dimensions and Assouad spectra, we show that, given a compact subset $E$ of a metric space, typically $R^n$, there is a convergent sequence of points contained in $E$ of the same dimension as $E$ itself. Moreover, under certain conditions it is possible for a sequence to witness the dimension of $E$ for many definitions of dimension simultaneously, for example in a self-affine set $E$ there is a single convergent sequence that has the same Assouad spectrum or intermediate dimension values as $E$ itself.
Comments18 pages
