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April 2017, 10(2): 213-240. doi: 10.3934/dcdss.2017011
A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy
Kristin Dettmers 1, , Robert Giza 2, , Rafael Morales 2, , John A. Rock 2,, and Christina Knox 3,
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Department of Mathematics and Statistics, California State Polytechnic University, Pomona, CA 91768, USA
Department of Mathematics, University of California, Riverside, CA 92521, USA
* Corresponding author:John A. Rock.
Received October 2015 Revised November 2016 Published January 2017
Fund Project: The authors Dettmers, Giza, and Rock were supported by NSF grant DMS-1247679.
Figure(3)
The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.
Keywords: Fractals, lattice, nonlattice, self-similarity, Hausdorff metric, Diophantine approximation, box-counting function, box-counting dimension, Minkowski content, Minkowski dimension, fractal strings, zeta functions, complex dimensions.
Mathematics Subject Classification: Primary: 11M41, 28A12, 28A80; Secondary: 11J99, 28A75, 28C15, 32A10, 32A20, 37B10, 37C25, 40A05, 40A10.
Citation: Kristin Dettmers, Robert Giza, Rafael Morales, John A. Rock, Christina Knox. A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 213-240. doi: 10.3934/dcdss.2017011
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Figure 1. On the left, an approximation of the Sierpiński gasket $S_G$ discussed in Example 2.14. On the right, an approximation of the Quarter Fractal $Q$ discussed in Example 2.15.
Figure 3. A lattice approximation of $\mathcal{D}_{\mathcal{L}_\phi}$, the complex dimensions of the Golden string $\Omega_\phi$ with lengths $\mathcal{L}_\phi$. The plots show the complex dimensions $\mathcal{D}_M=\{z \in{\mathbb{C}}: 2^{-z} + 2^{-z\phi_M} = 1\}$ where $\phi_M=f_{M+1}/f_M$ approximates $\phi$ for $M=2,\ldots,9$. In each case, the point $D$ denotes the Minkowski dimension of the approximating attractor, and the figure repeats with period $\mathbf{p}$. Note, however, that $\mathcal{D}_{\mathcal{L}_\phi}$ itself is not periodic. See Examples 2.13, 2.32, and 3.23 as well as Theorem 3.25 and Remark 3.28.
Figure 2. The geometric oscillations of the Cantor string $\Omega_{CS}$ seen in the plot and semilog plot of $N_{CS}(x)/x^{D}$, on the left and right, respectively. Here, $N_{CS}$ is the geometric counting function of the Cantor string and $D=\dim_BC=\log_32$. See Example 3.18. (The function is discontinuous, the vertical line segments are an artifact of the program used to generate the images.)
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Kristin Dettmers Robert Giza Rafael Morales John A. Rock Christina Knox