The Regular n-Flake Dust in ℝ2 Is Not Minkowski Measurable

Abstract

The Minkowski measurability of self-similar sets is closely linked to the lattice or non-lattice nature of the contraction ratios of the underlying iterated function system. A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. While the conjecture has been confirmed for self-similar subsets of R, the lattice case remains open in general in higher dimensions. This paper establishes non-Minkowski measurability for a class of regular n-flake dusts in R2. The proof uses hole points to show that the parallel volume fails to be twice differentiable at a critical radius, ruling out constancy of the associated log-periodic normalized volume. These results provide further support for the Lapidus conjecture.