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We show that any nontrivial self-similar subset of the real line that isinvariant under a lattice iterated function system (IFS) satisfying the openset condition (OSC) is not Minkowski measurable. So far, this was only knownfor special classes of such sets. Thereby, we provide the last puzzle-piece inproving that under OSC a nontrivial self-similar subset of the real line isMinkowski measurable iff it is invariant under a nonlattice IFS, a 25-year-oldconjecture.

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