Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

TL;DR

This paper develops a Minkowski measurability criterion for relative fractal drums in Euclidean spaces, using complex dimensions derived from fractal zeta functions, extending previous results from fractal strings to higher-dimensional sets.

Contribution

It generalizes Minkowski measurability criteria to a broad class of relative fractal drums using complex dimensions and zeta functions, expanding the scope beyond fractal strings.

Findings

Established a criterion linking complex dimensions to Minkowski measurability.

Connected gauge-Minkowski measurability to the nature and location of complex dimensions.

Provided examples illustrating the application of the criterion.

Abstract

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.