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

TL;DR

This paper proves that nontrivial lattice self-similar sets on the real line are not Minkowski measurable, completing a 25-year-old conjecture by characterizing Minkowski measurability in terms of lattice versus nonlattice invariance.

Contribution

It extends the understanding of Minkowski measurability by proving the non-measurability of all nontrivial lattice self-similar sets on the real line under OSC.

Findings

Lattice self-similar sets are not Minkowski measurable.

The result completes the characterization of Minkowski measurability for self-similar sets.

It confirms the conjecture that Minkowski measurability corresponds to nonlattice invariance.

Abstract

We show that any nontrivial 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 was only known for special classes of such sets. Thereby, we provide the last puzzle-piece in proving that under OSC a nontrivial self-similar subset of the real line is Minkowski measurable iff it is invariant under a nonlattice IFS, a 25-year-old conjecture.