欧几里得空间中紧集与相对分形鼓的 Minkowski 可测性准则
Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
AI总结：
本文利用相对分形鼓的分形 ζ 函数及其主复维数，建立任意维欧几里得空间中 RFD 的 Minkowski 可测性准则，并推广分形弦的经典结论。
摘要：
我们针对任意维数欧几里得空间中的一大类相对分形鼓（relative fractal drums，简称 RFD），依据其复维数建立了一项 Minkowski 可测性准则；这些复维数被定义为与其相关的分形 ζ 函数的极点。相对分形鼓是对欧几里得空间有界子集以及第一作者与其合作者广泛研究的分形弦的一种意义深远的推广。事实上，这里建立的 Minkowski 可测性准则是对第一作者与 M. van Frankenhuijsen 针对分形弦所得相应准则的推广。与分形弦的情形类似，这里建立的准则是根据所考虑相对鼓对应的主复维数的位置来表述的。这些复维数被定义为相应距离（或管）ζ 函数的极点，或更一般地，奇点。我们还思考了 RFD 的规范 Minkowski 可测性（gauge-Minkowski measurability）概念，并建立了若干将其与复维数的性质和位置联系起来的结果。（当底层标度不遵循经典幂律时，这一点尤其有用。）我们通过若干有趣的例子说明了所得结果及其应用。
英文摘要：
We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their associated fractal zeta functions. Relative fractal drums represent a far-reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings studied extensively by the first author and his collaborators. In fact, the Minkowski measurability criterion established here is a generalization of the corresponding one obtained for fractal strings by the first author and M.\ van Frankenhuijsen. Similarly as in the case of fractal strings, the criterion established here is formulated in terms of the locations of the principal complex dimensions associated with the relative drum under consideration. These complex dimensions are defined as poles or, more generally, singularities of the corresponding distance (or tube) zeta function. We also reflect on the notion of gauge-Minkowski measurability of RFDs and establish several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples.
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