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

TL;DR

This paper investigates the conditions under which self-similar fractals in Euclidean space are Minkowski measurable, confirming a conjecture that lattice type fractals are not measurable and providing explicit formulas in the non-lattice case.

Contribution

It proves the conjecture that lattice type fractals are not Minkowski measurable and derives explicit formulas for the content in the non-lattice case using the residue of a related zeta function.

Findings

Lattice type fractals are not Minkowski measurable.

Non-lattice fractals are Minkowski measurable with content expressed via zeta function residues.

The conjecture by Gatzouras is confirmed under mild conditions.

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.