*(English)*Zbl 0838.28006

Let $F\subset {\mathbb{R}}^{n}$, the Lebesgue measure $V\left({F}_{\epsilon}\right)$ of the $\epsilon $-neighbourhood ${F}_{\epsilon}:=\{x\in {\mathbb{R}}^{n}:\text{dist}(x,F)\le \epsilon \}$ 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 $aV\left({F}_{\epsilon}\right)\le {\epsilon}^{n-d}\le bV\left({F}_{\epsilon}\right))$, 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 $\mathbb{R}$ 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 $|{I}_{n}|\ge |{I}_{n+1}\left|\right)$ 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 $\mathbb{R}$ 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.