Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators

Abstract

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

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

“…One can easily check, by using the scaling property of the distance zeta function, that λA is strongly d-languid, for any λ ≥ 2 √ 3 and with κ d := −1. Hence, we can apply Theorem 3.1 in order to obtain the following exact pointwise tube formula, valid for all t ∈ (0, 1/2 √ 3), and which coincides with the one obtained in [27,28,29] and also, in [7]:…”

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

“…We can also recover and extend the significantly more general fractal tube formulas obtained (for fractal sprays and self-similar tilings) in[LapPeWi1] and used, in particular, in[LapPeWi2].…”

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).…”