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arXiv · 2607.00690

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

Uta Freiberg·Jonas Lippold
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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.

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Uta Freiberg, Jonas Lippold. 2026-07-01. The regular n-flake dust in $\mathbb{R}^2$ is not Minkowski measurable. https://arxiv.org/abs/2607.00690

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