Distance and tube zeta functions of fractals and arbitrary compact sets

TL;DR

This paper introduces new zeta functions for arbitrary compact sets in Euclidean space, linking their complex dimensions to Minkowski content and exploring properties of fractal and quasiperiodic sets.

Contribution

It extends the concept of zeta functions to arbitrary sets, connecting their analytic properties with geometric measures like Minkowski content.

Findings

The abscissa of convergence equals the upper box dimension.

Residues of the tube zeta function relate to Minkowski content.

Constructs examples of transcendentally quasiperiodic sets.

Abstract

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}\,\mathrm{d} x$ for all $s\in\mathbb{C}$ with $\operatorname{Re}\,s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_{\delta}$ is the $\delta$-neighborhood of $A$, where $\delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $\zeta_A$ is equal to $\overline\dim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $\zeta_A$ located on the critical line $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided $\zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $\tilde\zeta_A(s)=\int_0^{\delta} t^{s-N-1}|A_t|\,\mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $\tilde\zeta_A$ computed at $D=\dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers.