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

TL;DR

This paper develops fractal tube formulas for compact sets in Euclidean spaces using complex analysis and applies these to establish a new criterion for Minkowski measurability, extending previous work on fractal strings.

Contribution

It generalizes fractal tube formulas and Minkowski measurability criteria from fractal strings to higher-dimensional compact sets using complex dimensions.

Findings

Derived pointwise and distributional fractal tube formulas.

Established a new Minkowski measurability criterion.

Extended previous results from fractal strings to higher dimensions.

Abstract

We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.