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A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

  • * Corresponding author:John A. Rock.

    * Corresponding author:John A. Rock. 

The authors Dettmers, Giza, and Rock were supported by NSF grant DMS-1247679

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  • 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.

    Mathematics Subject Classification: Primary: 11M41, 28A12, 28A80; Secondary: 11J99, 28A75, 28C15, 32A10, 32A20, 37B10, 37C25, 40A05, 40A10.

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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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