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On the Minkowski measurability of fractals. (English) Zbl 0838.28006
Let \(F\subset \mathbb{R}^n\), the Lebesgue measure \(V(F_\varepsilon)\) of the \(\varepsilon\)-neighbourhood \(F_\varepsilon:= \{x\in \mathbb{R}^n: \text{dist} (x,F) \leq\varepsilon\}\) may be used to define the Minkowski dimension of \(F\). In particular, if \(V(F_\varepsilon) \approx \varepsilon^{n-d}\) as \(\varepsilon\to 0\) (i.e., for positive constants \(a\), \(b\) and for sufficiently small \(\varepsilon\) we have \(aV(F_\varepsilon)\leq \varepsilon^{n-d}\leq bV( F_\varepsilon))\), then the Minkowski dimension equals \(d\). In case \(V(F_\varepsilon)\sim \varepsilon^{n-d}\) (i.e., for some positive constant \(c\), \(V(F_\varepsilon)/ \varepsilon^{n-d}\to c\) as \(\varepsilon\to 0\)) we say that \(F\) is \(d\)-dimensional Minkowski measurable, with Minkowski constant \(c\). A complete characterization of Minkowski measurable compact subsets of \(\mathbb{R}\) was given by M. L. Lapidus and C. Pomerance [Proc. Lond. Math. Soc., III. Ser. 66, No. 1, 41-69 (1993; Zbl 0788.34083)]. This characterization states that the compact set \(F= I\setminus\cup I_n\) (\(I\) is a bounded closed interval with disjoint open subintervals \(I_n\) satisfying \(|I_n |\geq|I_{n+1} |)\) is Minkowski measurable if and only if \(|I_n|\sim cn^{-1/d}\) as \(n\to \infty\).
The author in this interesting work gives a rather simple proof of this characterization using dynamical systems arguments. He also examines selfsimilar subsets \(F\) of \(\mathbb{R}\) showing that, under some weak conditions on the ratios and gaps of the construction maps, \(F\) is Minkowski measurable. We should note here that the author uses some renewal theory arguments developed by S. Lalley [Acta Math. 163, No. 1/2, 1-55 (1989; Zbl 0701.58021)]. One should also pay notice to the class of Minkowski measurable fractals being closely related to the general problem of the Weyl-Berry conjecture on the distribution of eigenvalues of the Laplacian on domains with fractal boundaries.

28A80 Fractals
60K10 Applications of renewal theory (reliability, demand theory, etc.)
Full Text: DOI
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