On the Minkowski Measurability of Self-Similar Fractals in R^d

Abstract

M. Lapidus and C. Pomerance (1990-1993) and K.J. Falconer (1995) proved that a self-similar fractal in $\mathbb{R}$ is Minkowski-measurable iff it is of non-lattice type. D. Gatzouras (1999) proved that a self-similar fractal in $\mathbb{R}^d$ is Minkowski measurable if it is of non-lattice type (though the actual computation of the content is intractable with his approach) and conjectured that it is not Minkowski measurable if it is of lattice type. Under mild conditions we prove this conjecture and in the non-lattice case we improve his result in the sense that we express the content of the fractal in terms of the residue of the associated $\zeta$-function at the Minkowski-dimension.