Advertisement

Mathematische Zeitschrift

, Volume 283, Issue 3–4, pp 1049–1070 | Cite as

Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

  • Sabrina Kombrink
  • Erin P. J. Pearse
  • Steffen Winter
Article

Abstract

A long-standing conjecture of Lapidus states that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. It was shown by Falconer and Lapidus (working independently but both using renewal theory) that nonlattice self-similar subsets of \({\mathbb {R}}\) are Minkowski measurable, and the converse was shown by Lapidus and v. Frankenhuijsen a few years later, using complex dimensions. Around that time, Gatzouras used renewal theory to show that nonlattice self-similar subsets of \({\mathbb {R}}^d\) that satisfy the open set condition are Minkowski measurable for \(d \ge 1\). Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous work have been removed.

Keywords

Self-similar set Lattice and nonlattice case Minkowski dimension Minkowski measurability Minkowski content 

Mathematics Subject Classification

Primary 28A75 28A80 Secondary 28A12 

Notes

Acknowledgments

The first author was supported by Grant 03/113/08 of the Zentrale Forschungsförderung, Universität Bremen. The third author was supported by the Deutsche Forschungsgemeinschaft (DFG), Grant WI 3264/2-2.

References

  1. 1.
    Bandt, C., Hung, N.V., Rao, H.: On the open set condition for self-similar fractals. Proc. Am. Math. Soc. 134(5), 1369–1374 (2006). doi: 10.1090/S0002-9939-05-08300-0 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Demir, B., Deniz, A., Koçak, Ş., Üreyen, A.E.: Tube formulas for graph-directed fractals. Fractals 18(3), 349–361 (2010). doi: 10.1142/S0218348X10004919 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Deniz, A., Koçak, Ş., Özdemir, Y., Ratiu, A., Üreyen, A.E.: On the Minkowski measurability of self-similar fractals in \({\mathbb{R}}^d\). Turk. J. Math. 37(5), 830–846 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Deniz, A., Koçak, Ş., Özdemir, Y., Üreyen, A.E.: Tube formula for self-similar fractals with non-Steiner-like generators. In: Proceedings of the Gökova Geometry-Topology Conference 2012, pp. 123–145. International Press, Somerville (2013)Google Scholar
  5. 5.
    Deniz, A., Koçak, Ş., Özdemir, Y., Üreyen, A.E.: Tube formulas for self-similar and graph-directed fractals. Math. Intell. 35(3), 36–49 (2013). doi: 10.1007/s00283-013-9382-8 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deniz, A., Koçak, Ş., Özdemir, Y., Üreyen, A.E.: Tube volumes via functional equations. J. Geom. (2014). doi: 10.1007/s00022-014-0241-3 zbMATHGoogle Scholar
  7. 7.
    Falconer, K.J.: On the Minkowski measurability of fractals. Proc. Am. Math. Soc. 123(4), 1115–1124 (1995). doi: 10.2307/2160708 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Falconer, K.J.: Techniques in Fractal Geometry. Wiley, Chichester (1997)zbMATHGoogle Scholar
  9. 9.
    Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Hoboken (2003). doi: 10.1002/0470013850 CrossRefzbMATHGoogle Scholar
  10. 10.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. I, 3rd edn. Wiley, New York (1968)zbMATHGoogle Scholar
  11. 11.
    Gatzouras, D.: Lacunarity of self-similar and stochastically self-similar sets. Trans. Am. Math. Soc. 352(5), 1953–1983 (2000). doi: 10.1090/S0002-9947-99-02539-8 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981). doi: 10.1512/iumj.1981.30.30055 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kesseböhmer, M., Kombrink, S.: Minkowski content and fractal Euler characteristic for conformal graph directed systems. J. Fractal Geom. 2(2), 171–227 (2015). doi: 10.4171/JFG/19 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kneser, M.: Einige Bemerkungen über das Minkowskische Flächenmaß. Arch. Math. 6(5), 382–390 (1955). doi: 10.1007/BF01900510 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koçak, Ş., Ratiu, A.V.: Inner tube formulas for polytopes. Proc. Am. Math. Soc. 140(3), 999–1010 (2012). doi: 10.1090/S0002-9939-2011-11307-8 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kombrink, S.: Fractal curvature measures and Minkowski content for limit sets of conformal function systems. Ph.D. thesis, Universität Bremen. http://nbn-resolving.de/urn:nbn:de:gbv:46-00102477-19 (2011)
  17. 17.
    Lalley, S.P.: The packing and covering functions of some self-similar fractals. Indiana Univ. Math. J. 37(3), 699–710 (1988). doi: 10.1512/iumj.1988.37.37034 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lalley, S.P.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math. 163(1–2), 1–55 (1989). doi: 10.1007/BF02392732 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lapidus, M.L.: Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl-Berry conjecture. In: Ordinary and Partial Differential Equations, vol. IV (Dundee, 1992), Pitman Research Notes in Mathematics Series, vol. 289, pp. 126–209. Longman Science and Technology, Harlow (1993)Google Scholar
  20. 20.
    Lapidus, M.L., van Frankenhuijsen, M.: Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings, 2nd edn. Springer Monographs in Mathematics. Springer, New York (2013). doi: 10.1007/978-1-4614-2176-4 CrossRefzbMATHGoogle Scholar
  21. 21.
    Lapidus, M.L., Pearse, E.P.J.: Tube formulas for self-similar fractals. In: Analysis on Graphs and its Applications, Proceedings of Symposia in Pure Mathematics, vol. 77, pp. 211–230. American Mathematical Society, Providence. arXiv:0711.0173 (2008)
  22. 22.
    Lapidus, M.L., Pearse, E.P.J.: Tube formulas and complex dimensions of self-similar tilings. Acta Appl. Math. 112, 91–137 (2010). doi: 10.1007/s10440-010-9562-x MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lapidus, M.L., Pearse, E.P.J., Winter, S.: Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators. Adv. Math. 227, 1349–1398 (2011). doi: 10.1016/j.aim.2011.03.004 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lapidus, M.L., Pearse, E.P.J., Winter, S.: Minkowski measurability results for self-similar tilings and fractals with monophase generators. In: Carfi, D., Lapidus, M.L., Pearse, E.P.J., Van Frankenhuysen, M. (eds.) Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics. Contemporary Mathematics, vol. 600, pp. 185–203. American Mathematical Society Providence, Rhode Island (2013). doi: 10.1090/conm/600/11951
  25. 25.
    Pearse, E.P.J.: Complex dimensions of self-similar systems. Ph.D. thesis, University of California, Riverside (2006)Google Scholar
  26. 26.
    Pearse, E.P.J.: Canonical self-affine tilings by iterated function systems. Indiana Univ. Math J. 56(6), 3151–3169 (2007). doi: 10.1512/iumj.2007.56.3220 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pearse, E.P.J., Winter, S.: Geometry of canonical self-similar tilings. Rocky Mt. J. Math. 42(4), 1327–1357 (2012). doi: 10.1216/RMJ-2012-42-4-1327 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rataj, J., Winter, S.: Characterization of Minkowski measurability in terms of surface area. J. Math. Anal. Appl. 400(1), 120–132 (2013). doi: 10.1016/j.jmaa.2012.10.059 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Resman, M.: Invariance of the normalized Minkowski content with respect to the ambient space. Chaos Solitons Fractals 57, 123–128 (2013). doi: 10.1016/j.chaos.2013.10.001 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Schief, A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122(1), 111–115 (1994). doi: 10.2307/2160849 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Winter, S.: Curvature measures and fractals. Dissertationes Math. 453, 1–66 (2008). doi: 10.4064/dm453-0-1. (Rozprawy Mat.)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Winter, S.: Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets. Adv. Math. 274, 285–322 (2015). doi: 10.1016/j.aim.2015.01.005 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Sabrina Kombrink
    • 1
  • Erin P. J. Pearse
    • 2
  • Steffen Winter
    • 3
  1. 1.Institut für MathematikUniversität zu LübeckLübeckGermany
  2. 2.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  3. 3.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations