On the Minkowski measurability of fractals

Abstract

This note addresses two aspects of Minkowski measurability.First we present a short "dynamical systems" proof of the characterization of Minkowski measurable compact subsets of R .Second, we use a renewal theory argument to point out that "most" self-similar fractals are Minkowski measure able and calculate their Minkowski content.a similar notation for t -> oo .)Smooth or rectifiable curves are 1-dimensional Minkowski measurable, smooth surfaces are 2-dimensional Minkowski measurable, and so on (see Fdrer [4, 3.2.29]).Here we investigate conditions for fractal sets to be Minkowski measurable.It is possible to give a complete characterization of Minkowski measurable compact subsets of R.This was done by Lapidus and Pomerance [12,13] in their treatment of the 1-dimensional Weyl-Berry conjecture.The lengths of the complementary intervals to F are crucial.Let / be a bounded closed interval, with /" disjoint open subintervals of / satisfying |/"| > \In+x\ ( = 1,2,...) and |/| = Y^n=\ l^nl-(1^1 denotes the length of the interval 7.) Then the compact set F = I\\JnxLxIn is ^-dimensional Minkowski measurable if and