Fractal Zeta Functions and Fractal Drums

Distance and tube zeta functions of fractals in Euclidean spaces can be considered as a bridge between the geometry of fractal sets and the theory of holomorphic functions. This is first seen from their fundamental property: the upper box dimension of any bounded fractal is equal to the abscissa of convergence of its distance and tube zeta functions. Furthermore, under some natural conditions, the residue of the tube zeta function of a fractal, evaluated at its abscissa of convergence, is equal to its Minkowski content, a fractal analog of its volume. It is possible to obtain very general results dealing with the problem of meromorophic continuation of these two fractal zeta functions. We show, in particular, that the distance zeta function and the tube zeta function contain essentially the same information, both from the point of view of their meromorphic continuation (when it exists) to a given domain of the complex plane, of their poles (called visible complex dimensions) and their residues (or, more generally, their principal parts), which are related in a simple manner. Consequently, the higher-dimensional theory of complex dimensions can be developed by using either of these two fractal zeta functions, and much preferably, both of them since one of these zeta functions is often more natural or simpler to use in a given situation or example. A variety of examples are studied from this point of view throughout the book (including in this chapter, the (N − 1)-dimensional sphere, generalized Cantor sets and the a-string, and in later chapters, the N-dimensional analogs of the Sierpiński carpet and the Sierpiński gasket, as well as fractal nests, self-similar fractal sprays, two-parameter generalized Cantor sets, discrete and continuous spirals, geometric chirps, etc.). In the one-dimensional case (that is, in the case of fractal strings), we show that the geometric zeta function of a fractal string and the corresponding distance zeta function are equivalent (in a suitable sense), and, in fact, define the same complex dimensions (except possibly at s = 0); in particular, they have the same abscissa of convergence, equal to the upper Minkowski dimension of the fractal string (or, equivalently, of the associated fractal subset of the real line). As we shall see in later chapters, distance and tube zeta functions can also be viewed as a bridge to the transcendental theory of numbers. For these reasons, these new fractal zeta functions deserve to be seriously studied. In fact, as is suggested by the title of this research monograph, they are the central object of investigation in this chapter and, along with their poles (or ‘complex dimensions’), throughout the entire book.

In this chapter, we introduce the notion of relative fractal drums (or RFDs, in short). They represent a simple and natural extension of two fundamental objects of fractal analysis, simultaneously: that of bounded sets in \(\mathbb{R}^{N}\) (i.e., of fractals) and that of bounded fractal strings (introduced by the first author and Carl Pomerance in the early 1990s). Furthermore, there is a natural way to define their associated Minkowski contents and relative distance as well as tube zeta functions. We stress a new phenomenon exhibited by relative fractal drums: namely, their box dimensions can be negative as well (and even equal to −∞). This can be viewed as a property of their ‘flatness’, since it is related to the loss of the cone property. In short, a relative fractal drum (RFD) consists of an ordered pair (A, Ω), where A is an arbitrary (possibly unbounded) subset of \(\mathbb{R}^{N}\) and Ω is an open subset of \(\mathbb{R}^{N}\) of finite volume and such that Ω ⊆ A _δ , for some δ > 0. The corresponding zeta function, either a distance or tube zeta function, is denoted by ζ _{A, Ω} or \(\tilde{\zeta }_{A,\varOmega }\), respectively. We show that ζ _{A, Ω} and \(\tilde{\zeta }_{A,\varOmega }\) are connected via a key functional equation, which implies that their poles (i.e., the complex dimensions of the RFD (A, Ω)) are the same. We also extend to this general setting the main results of Chapters 2 and 3 concerning the holomorphicity and meromorphicity of the fractal zeta functions. We introduce the notion of transcendentally quasiperiodic relative fractal drums, using their tube functions. One way of constructing such drums is based on a carefully chosen sequence of generalized Cantor sets, as well as on the use of a classic result by Alan Baker from transcendental number theory. This construction and result extend the corresponding ones obtained in Chapter 3, in which we studied transcendentally quasiperiodic fractal sets. Furthermore, some explicit constructions of RFDs lead us naturally to introduce a new class of fractals, which we call hyperfractals. Particulary noteworthy among them are the maximal hyperfractals, for which the critical line \(\{\mathop{\mathrm{Re}}s = \overline{\dim }_{B}(A,\varOmega )\}\), where \(\overline{\dim }_{B}(A,\varOmega )\) is the relative upper box dimension of (A, Ω) and coincides with the abscissa of convergence of ζ _{A, Ω} or \(\tilde{\zeta }_{A,\varOmega }\), consists solely of nonisolated singularities of the corresponding fractal zeta function (i.e., of the relative distance or tube zeta function), ζ _{A, Ω} or \(\tilde{\zeta }_{A,\varOmega }\).

In this chapter, we reconstruct information about the geometry of relative fractal drums (and, consequently, compact sets) in \(\mathbb{R}^{N}\) from their associated fractal zeta functions. Roughly speaking, given a relative fractal drum (A, Ω) in \(\mathbb{R}^{N}\) (with N ≥ 1 arbitrary), we derive an asymptotic formula for its relative tube function t ↦ | A _t ∩Ω | as t → 0⁺, expressed as a sum taken over its complex dimensions of the residues of its (suitably modified and meromorphically extended) fractal zeta function. The resulting asymptotic formulas are called fractal tube formulas and are valid either pointwise or distributionally, as well as with or without an error term, depending on the growth properties of the associated fractal zeta functions. We note that these fractal tube formulas are expressed either in terms of the tube zeta function \(\tilde{\zeta }_{A,\varOmega }\) or, more interestingly, in terms of the distance zeta function ζ _{A, Ω} . The results of this chapter generalize to higher dimensions and arbitrary relative fractal drums the corresponding ones obtained previously for fractal strings by the first author and M. van Frankenhuijsen. We illustrate these results by obtaining fractal tube formulas for a number of well-known fractal sets, including the Sierpiński gasket and 3-carpet along with higher-dimensional analogs, a version of the graph of the Cantor function (i.e., of the devil’s staircase), fractal strings, fractal sprays, self-similar sprays and tilings, as well as certain non self-similar fractals, such as fractal nests and unbounded geometric chirps. We also apply these results in an essential way in order to obtain and establish a Minkowski measurability criterion for a large class of relative fractal drums (and, in particular, of bounded sets) in \(\mathbb{R}^{N}\), with N ≥ 1 arbitrary. More specifically, under appropriate hypotheses, a relative fractal drum (and, in particular, a bounded set) in \(\mathbb{R}^{N}\) of (upper) Minkowski dimension D is shown to be Minkowski measurable if and only if its only complex dimension with real part equal to D is D itself, and D is simple. We also discuss the notion of fractality defined in our context as the presence of at least one nonreal complex dimension. We show, in particular, that as is expected and intuitive, (a variant of) the Cantor graph (or devil’s staircase) is “fractal” in our sense, whereas as is well known, it is not “fractal” in Mandelbrots’s sense.