A family of non Minkowski measurable fractals in ℝ²

2025/06/03 by Freiberg, Uta, Lippold, Jonas
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2506.02640

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 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 ℝ² 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.

Citations

• Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

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