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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 kombrink@math.uni-luebeck.de
STEFFEN WINTER
Affiliation:
Karlsruhe Institute of Technology, Department of Mathematics, Englerstr. 2, 76131 Karlsruhe, Germany email steffen.winter@kit.edu
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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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