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In 2009, the first author introduced a class of zeta functions, called `distance zeta functions, which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions, defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of maximally hyperfractal compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of the N-dimensional Euclidean space).
The theory of zeta functions of fractal strings has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 20
In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions, associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of th
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 $d(x,A)$ denote
We study the essential singularities of geometric zeta functions $zeta_{mathcal L}$, associated with bounded fractal strings $mathcal L$. For any three prescribed real numbers $D_{infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{infty}<D_1le D$, we c
In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $mathcal F$ in a domain $Dsubset mathbb C,$ and for a positive const