MINKOWSKI CONTENT AND FRACTAL CURVATURES OF SELF-SIMILAR TILINGS AND GENERATOR FORMULAS FOR SELF-SIMILAR SETS

Abstract. We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corre-sponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curva-ture data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or Perron-Frobenius theory. Our approach based on the classical Renewal Theorem is much simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when a suitable volume function (or curvature functions, respectively) related to the generator are used. We also strengthen known re-sults on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case