Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

Abstract

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function ${\zeta }_A(s):={\int }_{A_{\delta }}d(x,A{)}^{s-N}\mathrm{d}x$, where $\delta >0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of ${\mathbb{R}}^N$. The abscissa of Lebesgue convergence $D({\zeta }_A)$ coincides with $D:={\overline{\dim }}_{B}A$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the "critical line" $\{\Re (s)=D\}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $t\to 0^{+}$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(\mathcal{M}+O(t^{\gamma }))$ as $t\to 0^{+}$, with $\gamma >0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(\log t^{-1})+O(t^{\gamma }))$ as $t\to 0^{+}$, where $G$ is a nonconstant periodic function and $\gamma >0$. In both cases, we show that ${\zeta }_A$ can be meromorphically extended (at least) to the open right half-plane $\{\Re (s)>D-\gamma \}$. Furthermore, up to a multiplicative constant, the residue of ${\zeta }_A$ evaluated at $s=D$ is shown to be equal to $\mathcal{M}$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line $\{\Re (s)=D\}$.

30 pages, 2 figures, improved parts of the paper and shortened the paper by reducing background material, to appear in Journal of mathematical analysis and applications in 2017

References in corpus (6)

• Distance and tube zeta functions of fractals and arbitrary compact sets
• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Fractal Strings and Multifractal Zeta Functions
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
• Fractality and Lapidus zeta functions at infinity

Cited by in corpus (7)

• Distance and tube zeta functions of fractals and arbitrary compact sets
• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
• Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions
• Fractal zeta functions of orbits of parabolic diffeomorphisms
• Fractal zeta functions at infinity and the $\varphi$-shell Minkowski content