Minkowski measurability results for self-similar tilings and fractals with monophase generators

Details Minkowski measurability results for self-similar tilings and fractals with monophase generators Minkowski measurability results for self-similar tilings and fractals with monophase generators

2011-04-08

arxiv.org/abs/1104.1641v1

Mathematics

Metric Geometry

12 pages, no figures

Scientific paper

In this appendix to the authors' paper [arXiv:1006.3807], we give conditions which characterize the 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 associated self-similar set (i.e., the fractal itself) is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. These results are all corollaries of the fractal tube formula(s) established in [arXiv:1006.3807] and [arXiv:math/0605527].

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