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.

15 pages, corrected typos, updated references, accepted for publication in Discrete and Continuous Dynamical Systems - Series S

References in corpus (5)

• Characterization of Minkowski measurability in terms of surface area
• Distance and tube zeta functions of fractals and arbitrary compact sets
• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

Cited by in corpus (6)

• Distance and tube zeta functions of fractals and arbitrary compact sets
• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Minkowski dimension and explicit tube formulas for $p$-adic fractal strings
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces