Positive Hausdorff dimensional spectrum for the critical almost Mathieu operator

TL;DR

This paper proves that for a dense set of frequencies with positive Hausdorff dimension, the spectrum of the critical almost Mathieu operator also has positive Hausdorff dimension, revealing complex spectral properties.

Contribution

It establishes the existence of a dense set of frequencies with positive Hausdorff dimension where the spectrum also has positive Hausdorff dimension, advancing understanding of spectral theory.

Findings

Existence of a dense set of frequencies with positive Hausdorff dimension.

Positive Hausdorff dimension of the spectrum for these frequencies.

Enhanced understanding of spectral properties of the critical almost Mathieu operator.

Abstract

We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.