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

Michel L. Lapidus; Goran Radunović; Darko Žubrinić

doi:10.3934/dcdss.2019007

Article Contents

2019, Volume 12, Issue 1: 105-117. Doi: 10.3934/dcdss.2019007

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

1.
Department of Mathematics, University of California, Riverside, CA 92521-0135, USA
2.
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia

^* Corresponding author
^* Corresponding author

Received: October 2016

Revised: May 2017

Published: July 2018

The research of Michel L. Lapidus was partially supported by the National Science Foundation under grants DMS-0707524 and DMS-1107750, as well as by the Institut des Hautes Études Scientifiques (IHÉS) where the first author was a visiting professor in the Spring of 2012 while part of this research was completed. The research of Goran Radunović and Darko Žubrinić was supported in part by the Croatian Science Foundation under the project IP-2014-09-2285 and by the Franco-Croatian PHC-COGITO project.

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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.

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Mathematics Subject Classification: Primary: 11M41, 28A12, 28A75, 28A80, 28B15, 42B20, 44A05; Secondary: 35P20, 40A10, 42B35, 44A10, 45Q05.

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