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Fractal Zeta Functions and Fractal Drums: Higher- Dimensional Theory of Complex Dimensions

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 ${; ; \mathbb R}; ; ^N$, where $N$ is any integer $\geq 1$. It is defined by $$\zeta_A(s)=int_{; ; A_{; ; \delta}; ; }; ; d(x, A)^{; ; s- N}; ; \D x, $$ where $d(x, A)$\label{; ; d(x, A)}; ; denotes the distance from $x$ to $A$ and $A_{; ; \delta}; ; $ is the $\delta$-neighborhood of~$A$. In this monograph, we investigate various properties of this ``distance zeta function". In particular, we prove that the zeta function is holomorphic in the half-plane $\{; ; {; ; \mathop{; ; \mathrm {; ; Re}; ; }; ; }; ; \, s>\overline\dim_BA\}; ; $, and that the bound $\overline\dim_BA$ is optimal. In other words, the abscissa of convergence of $\zeta_A$ is equal to $\overline\dim_BA$, which generalizes to arbitrary dimensions a well-known result for fractal strings (or equivalently, for arbitrary compact subsets of the real line $\eR$).\label{; ; eR}; ; Here, $\overline\dim_BA$ denotes the upper box (or Minkowski) dimension of~$A$. Extended to a meromorphic function $\zeta_A$, this ``distance zeta function" is shown to be an efficient tool for finding the box dimension of several new classes of subsets of ${; ; \mathbb R}; ; ^N$, like fractal nests, geometric chirps and multiple string chirps. It can also be used to develop a higher- dimensional theory of complex dimensions of arbitrary fractal sets in Euclidean spaces. For the sake of simplicity, we pay particular attention in this monograph 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\}; ; $. We also introduce a new zeta function, denoted by $\tilde\zeta_A$ and called a ``tube zeta function", and show, in particular, how to calculate the Minkowski content of a suitable (Minkowski measurable) bounded set $A$ in ${; ; \mathbb R}; ; ^N$ in terms of the residue of $\tilde\zeta_A(s)$ at $s=\dim_BA$, the box dimension of $A$. More generally, without assuming that $A$ is Minkowski measurable, we obtain analogous results, but now expressed as inequalities involving the upper and lower Minkowski contents of $A$. In addition, we obtain a new class of harmonic functions generated by fractal sets and represented via singular integrals. Furthermore, a class of sets is constructed with unequal upper and lower box dimensions, possessing alternating zeta functions. Moreover, by using a suitable notion of equivalence between zeta functions, we simplify some aspects of the theory of geometric zeta functions attached to fractal strings. In addition, we study the problem of the existence and constructing the meromorphic extensions of zeta functions of fractals ; in particular, we provide a natural sufficient condition for the existence of such extensions. An analogous problem is studied in the context of spectral zeta functions associated with bounded open subsets in Euclidean spaces with fractal boundary. We introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets with two parameters, along with the Gel'fond- Schneider theorem from the theory of transcendental numbers. With the help of this construction, we obtain an explicit example of a maximally hyperfractal set ; namely, a compact set $A\st\eR^N$ such that the associated distance and tube zeta functions have the critical line $\{; ; \re s=\ov\dim_BA\}; ; $ as a natural boundary. Actually, for this example, much more is true: every point of the critical line is a nonremovable singularity of the fractal zeta functions $\zeta_A$ and $\tilde\zeta_A$. Furthermore, we introduce the notion of relative fractal drum, which extends the notion of fractal string and of fractal drum. The associated definition of relative box dimension is such that it can achieve negative values as well. Using known results about the spectral asymptotics of fractal drums, and some of our earlier work, we recover known results about the existence of a (nontrivial) meromorphic extension of the spectral zeta function of a fractal drum. We also use some of our new results to establish the optimality of the upper bound obtained for the corresponding abscissa of meromorphic continuation of the spectral zeta function. Moreover, we develop a higher-dimensional theory of fractal tube formulas, with or without error terms, for relative fractal drums (and, in particular, for bounded sets) in $\eR^N$, for any $N\geq 1$. Such formulas, interpreted either pointwise or distributionally, enable us to express the volume of the tubular neighborhoods of the underlying fractal drums in terms of the associated complex dimensions. Therefore, they make apparent the deep connections between the theory of complex dimensions and the intrinsic oscillations of fractals. Accordingly, a geometric object is said to be ``fractal'' if it has at least one nonreal complex dimension (with positive real part) or else, the corresponding fractal zeta function has a natural boundary (along a suitable curve). We also formulate and establish a Minkowski measurability criterion for relative fractal drums (and, in particular, for bounded sets) in $\eR^N$, for any $N\geq 1$. More specifically, under suitable assumptions, a relative fractal drum (and, in particular, a bounded set) in $\eR^N$ is shown to be Minkowski measurable if and only if its only complex dimension with real part equal to its (upper) Minkowski dimension $D$ is $D$ itself, and it is simple. Throughout the book, we illustrate our results by a variety of examples, such as the Cantor set and string, the Cantor dust, a version of the Cantor graph (i.e., the ``devil's staircase''), fractal strings (including self- similar strings), fractal sprays (including self-similar sprays), the Sierpi\'nski gasket and carpet as well as their higher-dimensional counterparts, along with non self-similar examples, including fractal nests and geometric chirps. Finally, we propose a classification of bounded sets in Euclidean spaces, based on the properties of their tube functions (that is, the volume of their $\d$-neighborhoods, viewed as a function of the small positive number $\d$), and suggest various open problems concerning distance and tube zeta functions, along with their natural extensions in the context of ``relative fractal drums". Moreover, we suggest directions for future research in the higher-dimensional theory of the fractal complex dimensions of arbitrary compact subsets of Euclidean spaces (as well as more generally, of metric measure spaces). We stress that a significant advantage of the present theory of fractal zeta functions, and therefore, of the corresponding higher-dimensional theory of complex dimensions of fractal sets developed in this book, is that it is applicable to arbitrary bounded (or equivalently, compact) subsets of $\eR^N$, for any $N\ge1$. (At least in principle, it can also be extended to arbitrary compact metric measure spaces, although this is not explicitly done in this work.) In particular, no assumption of self- similarity or, more generally, of ``self- alikeness'' of any kind, is made about the underlying fractals (or, in the broader theory developed here, about the relative fractal drums under consideration).

zeta function ; geometric zeta function ; distance zeta function ; tube zeta function ; relative fractal drum ; fractal spray ; Minkowski content ; Minkowski measurable set ; gauge function

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