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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.

##### MSC:
 28A80 Fractals 60K10 Applications of renewal theory (reliability, demand theory, etc.)
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##### References:
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