Zeta Functions and Complex Dimensions of Relative Fractal Drums: Theory, Examples and Applications

TL;DR

This paper develops a comprehensive theory of zeta functions for relative fractal drums, extending previous fractal zeta function theories, and explores their complex dimensions, meromorphic extensions, and applications to fractal geometry.

Contribution

It introduces the concept of relative fractal drums and their associated distance zeta functions, unifying and extending prior fractal zeta function theories.

Findings

Construction of meromorphic extensions for RFD zeta functions

Introduction of transcendentally quasiperiodic RFDs with algebraically independent quasiperiods

Identification of maximal hyperfractals with singularities only on the critical line

Abstract

In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions', associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the theory of `geometric zeta functions' of bounded fractal strings. In this memoir, we introduce the class of `relative fractal drums' (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs, extend (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $\infty$-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called {\em maximal hyperfractals}, such that the critical line of (absolute) convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $m\ge1$, and even an infinite sequence of essential singularities along the critical line.