The regular n-flake dust in $\mathbb{R}^2$ is not Minkowski measurable

Uta Freiberg; Jonas Lippold

The regular n-flake dust in $\mathbb{R}^2$ is not Minkowski measurable

Metric Geometry 2026-07-01 v1

Authors: Uta Freiberg , Jonas Lippold

Abstract

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$ , the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents families of lattice-type subsets in $\mathbb{R}^2$ that are not Minkowski measurable, hence providing further support for the conjecture.

Keywords

Minkowski measurability self-similar sets lattice-type fractal dust Lapidus conjecture

Cite

@article{arxiv.2607.00690,
  title  = {The regular n-flake dust in $\mathbb{R}^2$ is not Minkowski measurable},
  author = {Uta Freiberg and Jonas Lippold},
  journal= {arXiv preprint arXiv:2607.00690},
  year   = {2026}
}