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

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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.

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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.

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Authors and Affiliations

  1. Institut für Mathematik, Universität zu Lübeck, Ratzeburger Allee 160, 23562, Lübeck, Germany

    Sabrina Kombrink

  2. Mathematics Department, California Polytechnic State University, San Luis Obispo, CA, 93407-0403, USA

    Erin P. J. Pearse

  3. Department of Mathematics, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    Steffen Winter

Authors
  1. Sabrina Kombrink
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  2. Erin P. J. Pearse
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  3. Steffen Winter
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Correspondence to Erin P. J. Pearse.

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Kombrink, S., Pearse, E.P.J. & Winter, S. Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable. Math. Z. 283, 1049–1070 (2016). https://doi.org/10.1007/s00209-016-1633-x

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  • DOI: https://doi.org/10.1007/s00209-016-1633-x

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