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Zbl 0838.28006
Falconer, K.J.
On the Minkowski measurability of fractals.
(English)
[J] Proc. Am. Math. Soc. 123, No.4, 1115-1124 (1995). ISSN 0002-9939; ISSN 1088-6826/e

Let $F\subset \bbfR^n$, the Lebesgue measure $V(F_\varepsilon)$ of the $\varepsilon$-neighbourhood $F_\varepsilon:= \{x\in \bbfR^n: \text {dist} (x,F) \leq\varepsilon\}$ 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 $\bbfR$ was given by {\it M. L. Lapidus} and {\it 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 $n\to \infty$. \par 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 $\bbfR$ 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 {\it 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.
[C.Karanikas (Thessaloniki)]
MSC 2000:
*28A80 Fractals
60K10 Appl. of renewal theory

Keywords: Minkowski measurability; Minkowski dimension; selfsimilar subsets; fractals; Weyl-Berry conjecture

Citations: Zbl 0788.34083; Zbl 0701.58021

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