Zeta functions and complex dimensions of relative fractal drums: theory, examples and applications

Abstract

relative fractal drum (RFD), fractal zeta functions, relative distance zeta function, relative tube zeta function, geometric zeta function of a fractal string, relative Minkowski content, relative Minkowski measurability, relative upper box (or Minkowski) dimension, relative complex dimensions of an RFD, holomorphic and meromorphic functions, abscissa of absolute and meromorphic convergence, transcendentally ∞-quasiperiodic function, transcendentally ∞-quasiperiodic RFD, a-string of higher order.

Fractal tube formulas for compact sets and relative fractal drums: oscillations, complex dimensions and fractality

“…(Indeed, since f (t) = 0 for all t ≥ δ and |A t ∩Ω| ≤ |Ω|, we easily see that t → t c−1 f (t) belongs to L 1 (0, +∞) for c ≥ N .) The stronger conclusion obtained in Theorem 2.19 requires the aforementioned results obtained in and [LapRaŽu4].…”

“…We note that RFDs having (even) principal complex dimensions of arbitrary orders exist and are relatively easy to construct, as was done in [LapRaŽu4,Section 4.4] and also in [LapRaŽu1, Subsection 4.2.2]. Furthermore, also in the just mentioned references, RFDs with principal complex dimensions of infinite order (i.e., with principal complex dimensions that are essential singularities of the associated fractal zeta function) have been constructed; see [LapRaŽu4,Section 4.4] and [LapRaŽu1, Subsection 4.2.2]. We stress that the theory of the present paper can also be applied if we allow complex dimensions of infinite order.…”

“…which is valid on any open connected subset U of C to which any of these two zeta functions has a meromorphic continuation (see [LapRaŽu4] or [LapRaŽu1]). This result is very useful since in many concrete examples, the distance zeta function is much easier to calculate than the tube zeta function.…”

“…Note that dim B (A, Ω) ∈ [−∞, N ]. We stress that the values of dim B (A, Ω) can indeed be negative, even equal to −∞; see [LapRaŽu4] or [LapRaŽu1, Chapter 4]. 9 Also note that for these definitions to make sense, it suffices that |A δ ∩ Ω| < ∞ for some δ > 0.…”

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

“…not to have a meromorphic continuation beyond that curve); such objects are now called hyperfractals in [42][43][44][45][46][47][48]. Recently, in [42][43][44][45], the authors have constructed maximally hyperfractal strings (and also compact subsets of R N , for any N ≥ 1), in the sense that the geometric (or fractal) zeta functions have singularities at every point of the vertical line {Re(s) = D}. (c) The mathematical theory of complex dimensions of fractal strings has many applications to a variety of subjects, including fractal geometry, spectral geometry, number theory, arithmetic geometry, geometric measure theory, probability theory, dynamical systems and mathematical physics (e.g.…”