Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Abstract

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 R N ).Keywords. Fractal set, fractal zeta functions, distance zeta function, tube zeta function, geometric zeta function of a fractal string, Minkowski content, Minkowski measurability, upper box (or Minkowski) dimension, complex dimensions of a fractal set, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, relative fractal drum, fractal tube formulas.

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

“…More precisely, we construct A by dividing the closed unit cube of R 3 into 27 congruent cubes and remove the open middle cube, then we iterate this step with each of the 26 remaining smaller closed cubes; and so on, ad infinitum. By choosing δ > 1/6, we deduce that ζ A is meromorphic on C and given for all s ∈ C by (see [33] or [40] In particular, P(ζA, C) = {0, 1, 2} ∪ log 3 26 + p Z , where p := 2π/ log 3; furthermore, each complex dimension of A is simple. Moreover, we have that res(ζ A ( · ; δ), 0) = 4π − 24 25 , res(ζ A ( · ; δ), 1) = 6π + 24 23 , res(ζ A ( · ; δ), 2) = 96 17 and, by letting ω k := log 3 26 + pk (for all k ∈ Z), res(ζ A ( · ; δ), ω k ) = 24 13 · 2 ω k ω k (ω k − 1)(ω k − 2) log 3 .…”

“…The development of the higher-dimensional theory of complex dimensions in [33,34,35,36,37,38,39,40] provides, among many other things, a new approach to the elusive notion of 'fractality'. To be more specific, for a set to be considered fractal, a commonly accepted proposal is that it should have a nontrivial fractal dimension, i.e., greater than its topological dimension; see, especially, [42], where the Hausdorff dimension was used.…”

“…In closing this introduction, we mention that works involving the notion of Minkowski measurability (or of Minkowski content) include [52,6,14,58,12,5,8,10,11,17,20,21,22,23,24,25,26,27,28,29,31,30,32,33,34,35,36,37,38,39,40,41,43,47,54], along with the many relevant references therein. References on tube formulas in the 'smooth setting' and in the 'fractal setting' are provided at the beginning of Section 3.…”

Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis

“…13], including a first attempt at a higher dimensional theory of complex dimensions for the special case of fractal sprays (in the sense of [30]) and self-similar tilings (see [7, namely, the 'distance zeta function' (introduced by the author in 2009) and the closely related 'tube zeta function'. For a comprehensive exposition of the resulting theory, we refer to the forthcoming book [42], along with the papers [43][44][45][46][47] and survey articles [48,75].…”

“…[42][43][44][45][46][47][48], the theory of complex dimensions of fractal strings developed in [5][6][7] has been extended to any bounded subset of Euclidean space R N (and even to 'relative fractal drums' of R N ) for any N ≥ 1, via the use of new 'fractal zeta functions'; namely, the 'distance zeta function' (introduced by the author in 2009) and the closely related 'tube zeta function'. For a comprehensive exposition of the resulting theory, we refer to the forthcoming book [42], along with the papers [43][44][45][46][47] and survey articles [48,75].…”