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

Sabrina Kombrink; Steffen Winter

doi:10.1017/etds.2018.26

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

Sabrina Kombrink, Steffen Winter

Mathematics

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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.

Original language	English
Pages (from-to)	221-232
Number of pages	12
Journal	Ergodic Theory and Dynamical Systems
Volume	40
Issue number	1
Early online date	10 Apr 2018
DOIs	https://doi.org/10.1017/etds.2018.26
Publication status	Published - Jan 2020

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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Cite this

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

Abstract

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