具有单相发生器的自相似镶嵌和分形的Minkowski可测性结果
Minkowski measurability results for self-similar tilings and fractals with monophase generators
摘要：
在以前的论文[arXiv：1006.3807]中，作者在相当一般的条件下获得了某些分形的管公式。在此基础上，我们给出了一类自相似镶嵌和自相似集的Minkowski可测性的一个刻画。在适当的假设下，简单的发电机（更准确地说，单相发电机）的自相似平铺被证明是Minkowski可测的，当且仅当相关的缩放zeta函数是非晶格类型。在平铺上的自然几何条件下，结果被转移到相关联的自相似集合（即，fracture本身）。此外，后者被证明是Minkowski可测的，当且仅当相关的缩放zeta函数是nonlattice类型。
英文摘要：
In a previous paper [arXiv:1006.3807], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.
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