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arXiv · 1501.03764

Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

Sabrina Kombrink·Erin P. J. Pearse·Steffen Winter
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Abstract

A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of $\mathbb{R}$ by Falconer, Lapidus and v.~Frankenhuijsen, and the forward direction was shown for fractal subsets of $\mathbb{R}^d$, $d \geq 2$, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.

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Sabrina Kombrink, Erin P. J. Pearse, Steffen Winter. 2015-01-15. Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable. https://doi.org/10.1007/s00209-016-1633-x

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