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

Abstract

We survey recent advances dealing with Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimensions 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 discussed 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 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 associated. We also discuss the notion of gauge-Minkowski measurability of RFDs and survey several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling law does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples.

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

“…These results extend to higher dimensions the corresponding tube formulas and Minkowski measurability criterion obtained for fractal strings in [41], §8.1 and §8.3, respectively. We point out that the detailed proofs of our main results (stated in a much more general form and within the broader context of relative fractal drums) can be found in the long papers corresponding to this article, [38,37]. Moreover, we note that in light of the functional equation (2.3), Theorems 3.1, 3.2 and 3.4 have an obvious analog for tube (instead of distance) zeta functions.…”

“…We will now introduce the notions of distance and tube zeta functions of compact sets and state their basic properties. These definitions have enabled us in [34,35,36,37,38,33,39] to develop a higher-dimensional extension of the theory of complex dimensions of fractal strings ( [41]), valid for arbitrary compact sets. Definition 2.1 (Fractal zeta functions, [34]).…”

“…The development of the higher-dimensional theory of complex dimensions in [33,34,35,36,37,38,39,40] provides, among many other things, a new approach to the elusive notion of 'fractality'. To be more specific, for a set to be considered fractal, a commonly accepted proposal is that it should have a nontrivial fractal dimension, i.e., greater than its topological dimension; see, especially, [42], where the Hausdorff dimension was used.…”

“…This latter part of the theory of complex dimensions is announced in the present article, along with sketches of proofs of the main results and illustrations of the results in two examples. The full proofs, and in the much greater generality of relative fractal drums, are provided in two articles [37,38] and can also be found in the research monograph [33,Ch. 5].…”

“…In closing this introduction, we mention that works involving the notion of Minkowski measurability (or of Minkowski content) include [52,6,14,58,12,5,8,10,11,17,20,21,22,23,24,25,26,27,28,29,31,30,32,33,34,35,36,37,38,39,40,41,43,47,54], along with the many relevant references therein. References on tube formulas in the 'smooth setting' and in the 'fractal setting' are provided at the beginning of Section 3.…”