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

TL;DR

This survey reviews the theory of complex dimensions in fractal geometry, highlighting its role in Minkowski and box-counting measurability, and extends the framework to self-similar sets in Euclidean spaces.

Contribution

It provides a comprehensive overview of complex dimensions related to fractal measurability and introduces an extension of the theory to higher-dimensional self-similar sets.

Findings

Complex dimensions characterize Minkowski and box-counting measurability.

Self-similar sets are Minkowski measurable if and only if they are nonlattice.

The theory extends to subsets of Euclidean space under mild separation conditions.

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.