Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable (Q303809)

A long-standing conjecture of Lapidus states that, under certain conditions, self-similar fractal sets fail to be self-similar if and only if they are of lattice type. It was proved in the mid-nineties by \textit{M. L. Lapidus} [in: Ordinary and partial differential equations. Volume IV. Proceedings of the twelfth Dundee conference held at the University of Dundee, UK, June 22-26, 1992. In honour of Professor D. S. Jones. Harlow: Longman Scientific \& Technical. 126--209 (1993; Zbl 0830.35094)], and independently by Falconer, that non-lattice self-similar subsets of \(\mathbb R\) are Minkowski measurable, and the converse was proved by Lapidus and Frankenhuijsen a few years later. Both proofs were based on renewal theory. In [Trans. Am. Math. Soc. 352, No. 5, 1953--1983 (2000; Zbl 0946.28006)], \textit{D. Gatzouras} used renewal theory to show that non-lattice self-similar subsets of \(\mathbb R^d\) that satisfy the open set condition are Minkowski measurable for \(d\geq 1\). The proof of the converse has been a much sought-after problem. In this paper, the authors prove a partial converse to the result of Gatzouras. While again based on renewal theory, the argument of the authors allows them to recover several previous results of this nature in a more efficient manner, and with greater generality.