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arXiv · 2506.02640

A family of non Minkowski measurable fractals in $\mathbb{R}^2$

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 a family of lattice-type subsets in $\mathbb{R}^2$ that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results.

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Uta Freiberg, Jonas Lippold. 2025-06-03. A family of non Minkowski measurable fractals in $\mathbb{R}^2$. https://arxiv.org/abs/2506.02640

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