Abstract
In this chapter, we show that some fundamental geometric and number-theoretic properties of fractals can be studied by using their distance and tube zeta functions. This will motivate us, in particular, to introduce several new classes of fractals. Especially interesting among them are the transcendentally quasiperiodic sets, since they can be placed at the crossroad between geometry and number theory. We shall need two deep results from transcendental number theory; namely, the theorem of Gel’fond–Schneider, and its extension due to Baker. In this context, the connecting link between the number theory and the geometry of fractals will be their tube zeta functions. A natural extension of the notion of distance zeta function leads us to introducing a general class of weighted zeta functions. Here, we introduce the space L ∞)(Ω): = ∩ p > 1 L p(Ω), called the limit L ∞-space, from which the weight functions are taken. Intuitively, a given weight function w from the space L ∞)(Ω) may only have very mild singularities, say, of logarithmic type. However, the set of singularities may be large, in the sense that its Hausdorff dimension can be arbitrarily close (and even equal) to N. A typical example is the function w(x) = logd(x, A) which appears under the integral sign when we differentiate the distance zeta function. We illustrate the efficiency of the use of distance zeta functions by computing the upper box dimension of several new classes of geometric objects, including geometric chirps, fractal nests and string chirps. These sets are closely related to bounded spirals and chirps in the plane. We also recall the construction of a class of fractals, called zigzagging fractals, for which the upper and lower box dimensions do not coincide, and show that the associated fractal zeta functions are alternating, in a suitable sense.
Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.
[No one shall expel us from the paradise that Cantor has created for us.]
David Hilbert (1862–1943)
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- 1.
The general definition of the oscillatory amplitude for a class a fractal sets in \(\mathbb{R}^{N}\) can be found on page 541.
- 2.
More generally, it suffices to assume that the Cantor sets A 1 and A 2 are contained in two compact unit intervals with disjoint interiors, respectively.
- 3.
According to Proposition 3.1.2 , this can be easily arranged.
- 4.
See footnote 57 on page 144.
- 5.
More generally, it suffices to assume that the interiors of the closed unit intervals I j are pairwise disjoint for j = 1,…,n.
- 6.
Since | A t | = | A t ∩ (0, a 1) | + 2t, we consider the tube function t ↦ | A t ∩ (0, a 1) | for convenience only, instead of t ↦ | A t | .
- 7.
The union of a countable family of fractal strings is introduced in Definition 4.5.11.
- 8.
More precisely, for this choice of δ, A δ is equal to the δ-neighborhood of the unit square [0, 1] 2.
- 9.
Indeed, the union of the deleted open squares in [0, 1]2, obtained during the construction of the Sierpiński carpet, is of 2-dimensional Lebesgue measure 1, since
$$\displaystyle{\sum _{k=1}^{\infty }8^{k-1}(3^{-k})^{2} = \frac{1} {9}\sum _{k=1}^{\infty }\Big(\frac{8} {9}\Big)^{k-1} = \frac{1} {9} \cdot \frac{1} {1 -\frac{8} {9}} = 1.}$$ - 10.
The complement (0, 1)2 ∖ A of the Sierpiński carpet A in (0, 1)2 can be thought of as the ‘dual Sierpiński carpet’ corresponding to the usual Sierpiński carpet A in the plane.
- 11.
More precisely, for this choice of δ, A δ is equal to the δ-neighborhood of the unit triangle \(\blacktriangle \).
- 12.
However, it does not seem to be known whether A is either Hausdorff or Minkowski nondegenerate, and in case it is degenerate, what is a corresponding gauge function h with respect to which it is h-Minkowski (or h-Hausdorff) nondegenerate. See Definition 6.1.4 and the discussion following it.
- 13.
For α > 1, the set A 1 is rectifiable (in other words, the sum of circumferences of all circles contained in A 1 is finite). We note that in the case when α = 1 we have dim B A 1 = 1, but the set is Minkowski degenerate. More precisely, its 1-dimensional Minkowski content exists and is equal to + ∞. However, it can be shown that h(t) = log(1∕t), for 0 < t < 1, is the corresponding gauge function of A 1, in the sense of Definition 6.1.4.
- 14.
- 15.
- 16.
Through the end of this discussion, (m k ) k = 1 ∞ is viewed as a decreasing sequence of positive real numbers with an associated sequence of generalized multiplicities denoted by \(\big(w_{k}^{(m)}\big)_{k=1}^{\infty }\).
- 17.
Here, for \(x \in \mathbb{R}\) , we have ⌊x⌋ = [x] (the integer part of x, also called the ‘floor’ of x).
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Lapidus, M.L., Radunović, G., Žubrinić, D. (2017). Applications of Distance and Tube Zeta Functions. In: Fractal Zeta Functions and Fractal Drums. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44706-3_3
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