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

Let F n , the Lebesgue measure V(F ε ) of the ε-neighbourhood F ε :={x n :dist(x,F)ε} may be used to define the Minkowski dimension of F. In particular, if V(F ε )ε n-d as ε0 (i.e., for positive constants a, b and for sufficiently small ε we have aV(F ε )ε n-d bV(F ε )), then the Minkowski dimension equals d. In case V(F ε )ε n-d (i.e., for some positive constant c, V(F ε )/ε n-d c as ε0) we say that F is d-dimensional Minkowski measurable, with Minkowski constant c. A complete characterization of Minkowski measurable compact subsets of 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=II n (I is a bounded closed interval with disjoint open subintervals I n satisfying |I n ||I n+1 |) is Minkowski measurable if and only if |I n |cn -1/d as n.

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 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:
28A80Fractals
60K10Applications of renewal theory