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

TL;DR

This paper advances the understanding of Minkowski measurability of self-similar fractals, showing that lattice-type sets with pluriphase generators generally fail to be Minkowski measurable, using renewal theory for a more general and simplified proof.

Contribution

It provides a partial converse to Lapidus's conjecture, extending previous results with a shorter proof that removes many technical conditions and applies to broader settings.

Findings

Lattice-type self-similar sets with pluriphase generators are generally not Minkowski measurable.

Renewal theory can be used to establish the non-measurability of these fractals.

The new proof simplifies previous approaches and broadens the applicable conditions.

Abstract

A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of $\mathbb{R}$ by Falconer, Lapidus and v.~Frankenhuijsen, and the forward direction was shown for fractal subsets of $\mathbb{R}^d$, $d \geq 2$, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.