Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions

TL;DR

This paper introduces a new class of fractal sets at infinity, studies their complex dimensions and zeta functions, and explores conditions for their meromorphic extension and Minkowski measurability, revealing quasiperiodic structures.

Contribution

It defines and analyzes fractal zeta functions at infinity, establishing their meromorphic extensions and revealing quasiperiodic complex dimensions with new examples.

Findings

Complex dimensions exhibit quasiperiodic structures, algebraic or transcendental.

Conditions for meromorphic extension of zeta functions are provided.

Existence of maximally hyperfractal sets with prescribed Minkowski dimension.

Abstract

In this paper we introduce an interesting family of relative fractal drums (RFDs in short) at infinity and study their complex dimensions which are defined as the poles of their associated Lapidus (distance) fractal zeta functions introduced in a previous work by the author. We define the tube zeta function at infinity and obtain a functional equation connecting it to the distance zeta function at infinity much as in the classical setting. Furthermore, under suitable assumptions, we provide general results about existence of meromorphic extensions of fractal zeta functions at infinity in the Minkowski measurable and nonmeasurable case. We also provide a sufficiency condition for Minkowski measurability as well as an upper bound for the upper Minkowski content, both in terms of the complex dimensions of the associated RFD. We show that complex dimensions of quasiperiodic sets at infinity posses a quasiperiodic structure which can be either algebraic or transcedental. Furthermore, we provide an example of a maximally hyperfractal set at infinity with prescribed Minkowski dimension, i.e., a set such that the abscissa of convergence of the corresponding fractal zeta function is in fact its natural boundary.