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

arXiv eprints
 Pub Date:
 November 2014
 arXiv:
 arXiv:1411.5733
 Bibcode:
 2014arXiv1411.5733L
 Keywords:

 Mathematical Physics;
 Mathematics  Complex Variables;
 Mathematics  Metric Geometry
 EPrint:
 15 pages, corrected typos, updated references, accepted for publication in Discrete and Continuous Dynamical Systems  Series S