# 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 and Continuous Dynamical Systems - S, 2019, 12 (1) : 105-117. doi: 10.3934/dcdss.2019007
##### 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. [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. [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. [4] W. Blaschke, Integralgeometrie, Chelsea, New York, 1949. [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. [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. [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. [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. [9] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.  doi: 10.1090/S0002-9947-1959-0110078-1. [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. [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. [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. [13] A. Gray, Tubes, 2nd edn., Progress in Math., vol. 221, Birkhäuser, Boston, 2004. doi: 10.1007/978-3-0348-7966-8. [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. [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. [16] D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Accademia Nazionale dei Lincei, Cambridge Univ. Press, Cambridge, 1997. [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. [18] J. Korevaar, Tauberian Theory: A Century of Developments, Springer-Verlag, Heidelberg, 2004. doi: 10.1007/978-3-662-10225-1. [19] O. Kowalski, Additive volume invariants of Riemannian manifolds, Acta Math., 145 (1980), 205-225.  doi: 10.1007/BF02414190. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [38] M. L. Lapidus, G. Radunović and D. Žubrinić, Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces, submitted for publication in the Proceedings of the 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals (June 2017), World Scientific, Singapore and London, 2019. (Also: e-print, arXiv: 1609.04498v2, [math-ph], 2018.) [39] M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, J. of Fixed Point Theory and Appl., 15 (2014), 321–378. Festschrift issue in honor of Haim Brezis' 70th birthday. doi: 10.1007/s11784-014-0207-y. [40] M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions: A general higher-dimensional theory, in: Fractal Geometry and Stochastics V (C. Bandt, K. Falconer and M. Zähle, eds.), Proc. Fifth Internat. Conf. (Tabarz, Germany, March 2014), Progress in Probability, vol. 70, Birkhäuser/Springer Internat., Basel, Boston and Berlin, 2015, 229–257. doi: 10.1007/978-3-319-18660-3_13. [41] M. L. Lapidus and M. van Frankenhuijsen, Fractality, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2nd rev. and enl. edn. (of the 2006 edn.), Springer Monographs in Mathematics, Springer, New York, 2006. doi: 10.1007/978-0-387-35208-4. [42] B. B. Mandelbrot, The Fractal Geometry of Nature, rev. and enl. edn. (of the 1977 edn.), W. H. Freeman, New York, 1983. [43] B. B. 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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. [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. [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. [4] W. Blaschke, Integralgeometrie, Chelsea, New York, 1949. [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. [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. [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. [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. [9] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.  doi: 10.1090/S0002-9947-1959-0110078-1. [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. [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. [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. [13] A. Gray, Tubes, 2nd edn., Progress in Math., vol. 221, Birkhäuser, Boston, 2004. doi: 10.1007/978-3-0348-7966-8. [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. [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. [16] D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Accademia Nazionale dei Lincei, Cambridge Univ. Press, Cambridge, 1997. [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. [18] J. Korevaar, Tauberian Theory: A Century of Developments, Springer-Verlag, Heidelberg, 2004. doi: 10.1007/978-3-662-10225-1. [19] O. Kowalski, Additive volume invariants of Riemannian manifolds, Acta Math., 145 (1980), 205-225.  doi: 10.1007/BF02414190. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [38] M. L. Lapidus, G. Radunović and D. Žubrinić, Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces, submitted for publication in the Proceedings of the 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals (June 2017), World Scientific, Singapore and London, 2019. (Also: e-print, arXiv: 1609.04498v2, [math-ph], 2018.) [39] M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, J. of Fixed Point Theory and Appl., 15 (2014), 321–378. Festschrift issue in honor of Haim Brezis' 70th birthday. doi: 10.1007/s11784-014-0207-y. [40] M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions: A general higher-dimensional theory, in: Fractal Geometry and Stochastics V (C. Bandt, K. Falconer and M. Zähle, eds.), Proc. Fifth Internat. Conf. (Tabarz, Germany, March 2014), Progress in Probability, vol. 70, Birkhäuser/Springer Internat., Basel, Boston and Berlin, 2015, 229–257. doi: 10.1007/978-3-319-18660-3_13. [41] M. L. Lapidus and M. van Frankenhuijsen, Fractality, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2nd rev. and enl. edn. (of the 2006 edn.), Springer Monographs in Mathematics, Springer, New York, 2006. doi: 10.1007/978-0-387-35208-4. [42] B. B. Mandelbrot, The Fractal Geometry of Nature, rev. and enl. edn. (of the 1977 edn.), W. H. Freeman, New York, 1983. [43] B. B. 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