A family of non Minkowski measurable fractals in $\mathbb{R}^2$

TL;DR

This paper constructs a new family of lattice-type fractals in two-dimensional space that are not Minkowski measurable, providing evidence for Lapidus's conjecture in higher dimensions.

Contribution

It introduces a novel family of non Minkowski measurable lattice-type fractals in $\

Findings

Identifies a new class of non Minkowski measurable fractals in $\

Supports the Lapidus conjecture in $\

Explains limitations of previous results in higher dimensions.

Abstract

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. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in $\mathbb{R}^2$ that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results.