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February 2019, 12(1): 105-117. doi: 10.3934/dcdss.2019007
Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces
Michel L. Lapidus 1,, , Goran Radunović 2, and Darko Žubrinić 2,
Department of Mathematics, University of California, Riverside, CA 92521-0135, USA
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
* Corresponding author
Received October 2016 Revised May 2017 Published July 2018
Fund Project: 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.
Keywords: Mellin transform, complex dimensions, relative fractal drum, fractal set, box dimension, fractal zeta function, distance zeta function, tube zeta function, fractal string, Minkowski content, Minkowski measurability criterion, Minkowski measurable set.
Mathematics Subject Classification: Primary: 11M41, 28A12, 28A75, 28A80, 28B15, 42B20, 44A05; Secondary: 35P20, 40A10, 42B35, 44A10, 45Q05.
Citation: Michel L. Lapidus, Goran Radunović, Darko Žubrinić. Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 105-117. doi: 10.3934/dcdss.2019007
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Michel L. Lapidus Goran Radunović Darko Žubrinić