# 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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##### 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. Pearse and M. van Frankenhuijsen (eds.), Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I, Fractals in Pure Mathematics, Contemporary Mathematics, vol. 600, Amer. Math. Soc., Providence, R. I., 2013.Google Scholar [7] A. Deniz, Ş. Koçak, Y. Özdemir and A. E. Üreyen, Tube volumes via functional equations, J. Geom., 106 (2015), 153-162. doi: 10.1007/s00022-014-0241-3. Google Scholar [8] K. J. Falconer, On the Minkowski measurability of fractals, Proc. Amer. Math. Soc., 123 (1995), 1115-1124. doi: 10.1090/S0002-9939-1995-1224615-4. Google Scholar [9] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. doi: 10.1090/S0002-9947-1959-0110078-1. Google Scholar [10] J. Fleckinger and D. Vassiliev, An example of a two-term asymptotics for the "counting function" of a fractal drum, Trans. Amer. Math. Soc., 337 (1993), 99-116. doi: 10.2307/2154311. Google Scholar [11] M. 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. Z., 246 (2004), 237-272. doi: 10.1007/s00209-003-0597-9. Google Scholar [16] D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Accademia Nazionale dei Lincei, Cambridge Univ. Press, Cambridge, 1997. Google Scholar [17] S. Kombrink, A survey on Minkowski measurability of self-similar sets and self-conformal fractals in $\mathbb{R}^d$, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics, I. Fractals in pure mathematics, 135–159, Contemp. Math., 600, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/conm/600/11931. Google Scholar [18] J. Korevaar, Tauberian Theory: A Century of Developments, Springer-Verlag, Heidelberg, 2004. doi: 10.1007/978-3-662-10225-1. Google Scholar [19] O. Kowalski, Additive volume invariants of Riemannian manifolds, Acta Math., 145 (1980), 205-225. doi: 10.1007/BF02414190. Google Scholar [20] M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc., 325 (1991), 465-529. doi: 10.1090/S0002-9947-1991-0994168-5. Google Scholar [21] M. L. Lapidus, Spectral and fractal geometry: From the Weyl–Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function, in: Differential Equations and Mathematical Physics (C. Bennewitz, ed.), Proc. Fourth UAB Internat. Conf. (Birmingham, March 1990), Academic Press, New York, 186 (1992), 151–181. doi: 10.1016/S0076-5392(08)63379-2. Google Scholar [22] M. L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl–Berry conjecture, in: Ordinary and Partial Differential Equations (B. D. Sleeman and R. J. Jarvis, eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Mathematics Series, vol. 289, Longman Scientific and Technical, London, 1993, 126–209. Google Scholar [23] M. L. Lapidus, In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes, research monograph, Amer. Math. Soc., Providence, R. I., 2008. doi: 10.1090/mbk/051. Google Scholar [24] M. L. Lapidus, The sound of fractal strings and the Riemann hypothesis, in: Analytic Number Theory: In Honor of Helmut Maier's 60th Birthday (C. B. Pomerance and T. Rassias, eds.), Springer Internat. Publ. Switzerland, Cham, 2015, 201–252. Google Scholar [25] M. L. Lapidus and H. Maier, Hypothèse de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiée, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 19-24. Google Scholar [26] M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. 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. 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. (Also: e-print, arXiv: 1506.03525v3, [math-ph], 2016; IHES preprint, IHES/M/15/15, 2015.) doi: 10.1016/j.aim.2016.11.034. Google Scholar [35] M. L. Lapidus, G. Radunović and D. Žubrinić, Complex dimensions of fractals and meromorphic extensions of fractal zeta functions, J. Math. Anal. Appl., 453 (2017), 458–484. (Also: e-print, arXiv: 1508.04784v4, [math-ph], 2016.) doi: 10.1016/j.jmaa.2017.03.059. Google Scholar [36] M. L. Lapidus, G. Radunović and D. Žubrinić, Zeta functions and complex dimensions of relative fractal drums: Theory, examples and applications, Dissertationes Math. (Rozprawy Mat.), 526 (2017), 1–105. (Also: e-print, arXiv: 1603.00946v3, [math-ph], 2016.) doi: 10.4064/dm757-4-2017. Google Scholar [37] M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal tube formulas for compact sets and relative fractal drums: Oscillations, complex dimensions and fractality, J. Fractal Geom., 5 (2018), 1–119. (DOI: 10.4171/JFG/57)(Also: e-print, arXiv:1604.08014v5, [math-ph], 2018.) doi: 10.4171/JFG/57. Google Scholar [38] M. L. Lapidus, G. Radunović and D. 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