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arXiv · 1510.06467

A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

Kristin Dettmers·Robert Giza·Christina Knox·Rafael Morales·John A. Rock
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Abstract

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.

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Kristin Dettmers, Robert Giza, Christina Knox, Rafael Morales, John A. Rock. 2015-12-08. A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy. https://arxiv.org/abs/1510.06467

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