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On the Minkowski measurability of fractals. (English) Zbl 0838.28006

Let $F\subset {ℝ}^{n}$, the Lebesgue measure $V\left({F}_{\epsilon }\right)$ of the $\epsilon$-neighbourhood ${F}_{\epsilon }:=\left\{x\in {ℝ}^{n}:\text{dist}\left(x,F\right)\le \epsilon \right\}$ may be used to define the Minkowski dimension of $F$. In particular, if $V\left({F}_{\epsilon }\right)\approx {\epsilon }^{n-d}$ as $\epsilon \to 0$ (i.e., for positive constants $a$, $b$ and for sufficiently small $\epsilon$ we have $aV\left({F}_{\epsilon }\right)\le {\epsilon }^{n-d}\le bV\left({F}_{\epsilon }\right)\right)$, then the Minkowski dimension equals $d$. In case $V\left({F}_{\epsilon }\right)\sim {\epsilon }^{n-d}$ (i.e., for some positive constant $c$, $V\left({F}_{\epsilon }\right)/{\epsilon }^{n-d}\to c$ as $\epsilon \to 0$) we say that $F$ is $d$-dimensional Minkowski measurable, with Minkowski constant $c$. A complete characterization of Minkowski measurable compact subsets of $ℝ$ 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 $|{I}_{n}|\ge |{I}_{n+1}|\right)$ is Minkowski measurable if and only if $|{I}_{n}|\sim c{n}^{-1/d}$ 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 $ℝ$ 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