Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

We consider the asymptotic behaviour of the volume of the Minkowski sausage, the counting function of the Dirichlet Laplacian, the partition function and the heat content for an iterated set script G sign ? R^m with fractal boundary. We show, using the renewal theory (well known in probability) that in all cases the asymptotic behaviour depends essentially on whether the set of logarithms of the similitudes used in the construction of script G sign is arithmetic.