Naslov
Distance and tube zeta functions of fractals and arbitrary compact sets

Autori
Lapidus, Michel L. ; Radunović, Goran ; Žubrinić, Darko

Izvornik
Advances in mathematics (0001-8708) 307 (2017); 1215-1267

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
zeta function ; distance zeta function ; tube zeta function ; fractal set ; fractal string ; box dimension ; principal complex dimensions ; Minkowski content ; Minkowski measurable set ; residue ; Dirichlet integral ; transcendentally quasiperiodic set.

Sažetak
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 $\zeta_A(s)=\int_{; ; ; ; ; A_{; ; ; ; ; \delta}; ; ; ; ; }; ; ; ; ; d(x, A)^{; ; ; ; ; s-N}; ; ; ; ; \D x$ for all $s\in\Ce$ with $\operatorname{; ; ; ; ; Re}; ; ; ; ; \, s$ sufficiently large, and we call it the {; ; ; ; ; \em distance zeta function}; ; ; ; ; of $A$. Here, $d(x, A)$\label{; ; ; ; ; d(x, A)}; ; ; ; ; denotes the Euclidean distance from $x$ to $A$ and $A_{; ; ; ; ; \delta}; ; ; ; ; $ is the $\delta$-neighborhood of~$A$, where $\d$ 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, $\tilde\zeta_A(s)=\int_0^\d t^{; ; ; ; ; s-N-1}; ; ; ; ; |A_t|\, \D t$, called the {; ; ; ; ; \em 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 {; ; ; ; ; \em 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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

Napomena
Preprint: https://arxiv.org/abs/1506.03525

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