Distance and tube zeta functions of fractals and arbitrary compact 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 $N\ge 1$. 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 $\mathrm{Re}s$ 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 ${\overline{\dim }}_{B}A$, 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 $\{\mathrm{Re}s={\overline{\dim }}_{B}A\}$, 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 $D={\dim }_{B}A$ (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.

54 pages, corrected misprints, reduced number of self-citations

References in corpus (7)

• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
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Cited by in corpus (7)

• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions
• Fractal zeta functions of orbits of parabolic diffeomorphisms
• Maximally singular solutions of Laplace equations
• Fractal zeta functions at infinity and the $\varphi$-shell Minkowski content