Tube Formulas and Complex Dimensions of Self-Similar Tilings

Abstract

Abstract. We use the self-similar tilings constructed in [32] to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in R d . The resulting power series in ε is a fractal extension of Steiner's classical tube formula for convex bodies K ⊆ R d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i = 0, 1, . . . , d − 1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in [30].

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

“…Many applications and extensions of fractal string theory and/or of the corresponding theory of complex fractal dimensions can be found throughout the books La5] and in [La1,La2,La3,La4,LaPo1,LaPo2,LaPo3,LaMa1,LaMa2,HeLa,HamLa,Tep,LaPe,LaPeWi,LaLeRo,ElLaMaRo,LaLu1,LaLu2,LalLa1,LalLa2,LaRaZu,HerLa1,HerLa2,HerLa3,HerLa4,La6]. These include, in particular, applications to various aspects of number theory and arithmetic geometry, dynamical systems, spectral geometry, geometric measure theory, noncommutative geometry, mathematical physics and nonarchimedean analysis.…”

Fractal Strings and Multifractal Zeta Functions

“…A useful guide in this endeavor should be provided by the recent work of Lapidus and Pearse on the complex dimensions of the Koch snowflake curve (see [21], as summarized in [29, Section 12.3.1]) and more generally but from a different point of view, on the zeta functions and complex dimensions of self-similar fractals and tilings in R d (see [22,23] and [46], as briefly described in [29,Section 12.3.2], along with the associated tube formulas).…”

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

“…We note that recent developments in the theory are described in [7, ch. 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, §13.1], based on [63][64][65][66]72]), p-adic fractal strings and associated non-Archimedean fractal tube formulae (see [7, §13.2], based on [56][57][58][59][60]), multi-fractal zeta functions and their 'tapestries' of complex dimensions (see [7, §13.3], based on [50,55,67]), random fractal strings (such as stochastically self-similar strings and the zero-set of Brownian motion) and their spectra (see [7, §13.4], based on [51]), as well as a new approach to the RH based on a conjectural Riemann flow of fractal membranes (i.e. quantized fractal strings) and correspondingly flows of zeta functions (or 'partition functions') and of the associated zeros (see [7, §13.5], which gives a brief overview of the aforementioned book [20], In search of the Riemann zeros).…”