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

Sabrina Kombrink, Steffen Winter

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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 languageEnglish
Pages (from-to)221-232
Number of pages12
JournalErgodic Theory and Dynamical Systems
Volume40
Issue number1
Early online date10 Apr 2018
DOIs
Publication statusPublished - Jan 2020

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