# American Institute of Mathematical Sciences

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

 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

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.

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.

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
##### References:
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Soc.(2), 52 (1995), 15-34.  doi: 10.1112/jlms/52.1.15.  Google Scholar [27] M. L. Lapidus and E. P. J. Pearse, Tube formulas and complex dimensions of self-similar tilings, Acta Applicandae Mathematicae, 112 (2010), 91-136.  doi: 10.1007/s10440-010-9562-x.  Google Scholar [28] M. L. Lapidus, E. P. J. Pearse and S. Winter, Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. in Math., 227 (2011), 1349-1398.  doi: 10.1016/j.aim.2011.03.004.  Google Scholar [29] M. L. Lapidus, E. P. J. Pearse and S. Winter, Minkowski measurability results for self-similar tilings and fractals with monophase generators, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics, I. Fractals in pure mathematics, 185–203, Contemp. Math., 600, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/conm/600/11951.  Google Scholar [30] M. L. Lapidus and C. Pomerance, Fonction zêta de Riemann et conjecture de Weyl-Berry pour les tambours fractals, C. R. 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##### References:
 [1] T. Bedford and A. M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc.(3), 64 (1992), 95-124.  doi: 10.1112/plms/s3-64.1.95.  Google Scholar [2] M. V. Berry, Distribution of modes in fractal resonators, Structural Stability in Physics (W. Güttinger and H. Eikemeier, eds.), pp. 51–53, Springer Ser. Synergetics, 4, Springer, Berlin, 1979. doi: 10.1007/978-3-642-67363-4_7.  Google Scholar [3] M. V. Berry, Some geometric aspects of wave motion: Wavefront dislocations, diffraction catastrophes, diffractals, Geometry of the Laplace Operator, Proc. Sympos. Pure Math., 36, Amer. Math. Soc., Providence, R. I., 1980, 13–38.  Google Scholar [4] W. Blaschke, Integralgeometrie, Chelsea, New York, 1949. Google Scholar [5] J. Brossard and R. Carmona, Can one hear the dimension of a fractal?, Commun. Math. Phys., 104 (1986), 103-122.  doi: 10.1007/BF01210795.  Google Scholar [6] D. Carfì, M. L. Lapidus, E. P. J. 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Frantz, Lacunarity, Minkowski content, and self-similar sets in $\mathbb{R}$, in: Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot (M. L. Lapidus and M. van Frankenhuijsen, eds.), Proc. Sympos. Pure Math., 72, Part 1, Amer. Math. Soc., Providence, R. I., 2004, 77–91.  Google Scholar [12] D. Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc., 352 (2000), 1953-1983.  doi: 10.1090/S0002-9947-99-02539-8.  Google Scholar [13] A. Gray, Tubes, 2nd edn., Progress in Math., vol. 221, Birkhäuser, Boston, 2004. doi: 10.1007/978-3-0348-7966-8.  Google Scholar [14] C. Q. He and M. L. Lapidus, Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function, Memoirs Amer. Math. Soc., 127 (1997), x+97 pp. doi: 10.1090/memo/0608.  Google Scholar [15] D. Hug, G. Last and W. Weil, A local Steiner-type formula for general closed sets and applications, Math. 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Acad. Sci. Paris Sér. I Math., 310 (1990), 343-348.   Google Scholar [31] M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc.(3), 66 (1993), 41-69.  doi: 10.1112/plms/s3-66.1.41.  Google Scholar [32] M. L. Lapidus and C. Pomerance, Counterexamples to the modified Weyl-Berry conjecture on fractal drums, Math. Proc. Cambridge Philos. Soc., 119 (1996), 167-178.  doi: 10.1017/S0305004100074053.  Google Scholar [33] M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, Springer Monographs in Mathematics, Springer, New York, 2017. doi: 10.1007/978-3-319-44706-3.  Google Scholar [34] M. L. Lapidus, G. Radunović and D. Žubrinić, Distance and tube zeta functions of fractals and arbitrary compact sets, Adv. in Math., 307 (2017), 1215–1267. 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