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

TL;DR

This paper characterizes Minkowski measurability of certain self-similar tilings and fractals using tube formulas, linking it to the lattice or nonlattice nature of their scaling zeta functions.

Contribution

It provides a new characterization of Minkowski measurability for self-similar tilings and sets based on the properties of their scaling zeta functions.

Findings

Minkowski measurability of tilings with monophase generators is equivalent to nonlattice scaling zeta functions.

The results extend to the associated self-similar sets under natural geometric conditions.

The paper generalizes previous tube formula results to characterize fractal measurability.

Abstract

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.