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

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.