Fractal zeta functions and complex dimensions: A general higher-dimensional theory

TL;DR

This paper develops a comprehensive higher-dimensional theory of fractal zeta functions, connecting them with complex dimensions, Minkowski content, and fractal geometry, including new classes of fractal sets with maximal hyperfractality.

Contribution

It introduces a general higher-dimensional framework for fractal zeta functions, including distance and tube zeta functions, and constructs maximally hyperfractal sets with singularities at all critical points.

Findings

Distance zeta function's abscissa matches upper box dimension.

Construction of maximally hyperfractal sets with singularities at every critical line point.

Established fractal tube formulas and Minkowski measurability criteria.

Abstract

In 2009, the first author introduced a class of zeta functions, called `distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of the N-dimensional Euclidean space).