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Lattice self-similar sets on the real line are not Minkowski measurable

Published online by Cambridge University Press:  10 April 2018

SABRINA KOMBRINK  and
STEFFEN WINTER
SABRINA KOMBRINK
Affiliation:
Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany email [email protected]
STEFFEN WINTER
Affiliation:
Karlsruhe Institute of Technology, Department of Mathematics, Englerstr. 2, 76131 Karlsruhe, Germany email [email protected]
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Abstract

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 221 - 232
Copyright
© Cambridge University Press, 2018 

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References

Bandt, C. and Mesing, M.. Self-affine fractals of finite type. Banach Center Publ. 84(1) (2009), 131–148.Google Scholar
Bandt, C., Mekhontsev, D. and Tetenov, A.. A single fractal pinwheel tile. Proc. Amer. Math. Soc. 146(3) (2018), 1271–1285.Google Scholar
Falconer, K. J.. On the Minkowski measurability of fractals. Proc. Amer. Math. Soc. 123(4) (1995), 1115–1124.Google Scholar
Gatzouras, D.. Lacunarity of self-similar and stochastically self-similar sets. Trans. Amer. Math. Soc. 352(5) (2000), 1953–1983.Google Scholar
He, X.-G. and Lau, K.-S.. On a generalized dimension of self-affine fractals. Math. Nachr. 281(8) (2008), 1142–1158.10.1002/mana.200510666Google Scholar
Kesseböhmer, M. and Kombrink, S.. Minkowski content and fractal Euler characteristic for conformal graph directed systems. J. Fractal Geom. 2(2) (2015), 171–227.10.4171/JFG/19Google Scholar
Kombrink, S., Pearse, E. P. J. and Winter, S.. Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable. Math. Z. 283(3–4) (2016), 1049–1070.10.1007/s00209-016-1633-xGoogle Scholar
Lapidus, M. L.. Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl–Berry conjecture. Ordinary and Partial Differential Equations, Vol. IV (Dundee 1992) (Pitman Research Notes in Mathematics Series, 289) . Longman Scientific & Technical, Harlow, 1993, pp. 126–209.Google Scholar
Lapidus, M. L. and Pomerance, C.. The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc. Lond. Math. Soc. (3) 66(1) (1993), 41–69.Google Scholar
Lapidus, M. L., Radunović, G. and Žubrinić, D.. Fractal Zeta Functions and Fractal Drums, Higher-Dimensional Theory of Complex Dimensions (Springer Monographs in Mathematics) . Springer, Cham, 2017.10.1007/978-3-319-44706-3Google Scholar
Lapidus, M. L. and van Frankenhuysen, M.. Complex dimensions of fractal strings and zeros of zeta functions. Fractal Geometry and Number Theory. Birkhäuser, Boston, 2000.Google Scholar
Lapidus, M. L. and van Frankenhuijsen, M.. Fractal Geometry, Complex Dimensions and Zeta Functions, Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics) , 2nd edn. Springer, New York, 2013.Google Scholar
Mandelbrot, B. B.. Measures of fractal lacunarity: Minkowski content and alternatives. Fractal Geometry and Stochastics (Finsterbergen 1994) (Progress in Probability, 37) . Birkhäuser, Basel, 1995, pp. 15–42.Google Scholar
Pearse, E. P. J. and Winter, S.. Geometry of canonical self-similar tilings. Rocky Mountain J. Math. 42(4) (2012), 1327–1357.10.1216/RMJ-2012-42-4-1327Google Scholar
Schief, A.. Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122(1) (1994), 111–115.Google Scholar
Stachó, L. L.. On the volume function of parallel sets. Acta Sci. Math. (Szeged) 38(3–4) (1976), 365–374.Google Scholar
Winter, S.. Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets. Adv. Math. 274 (2015), 285–322.10.1016/j.aim.2015.01.005Google Scholar