A classification of aperiodic order via spectral metrics and Jarn\'ik sets

TL;DR

This paper explores the relationship between spectral metrics, Diophantine properties, and complexity notions in Sturmian subshifts, establishing a connection with Jarník sets and calculating their Hausdorff dimension.

Contribution

It introduces new classifications of aperiodic order using spectral metrics and Diophantine properties, extending known complexity notions and linking them to geometric measure theory.

Findings

Hausdorff dimension of level sets is 2/(α+1)

Characterization of Sturmian subshifts via spectral regularity and Diophantine properties

Establishment of a connection between complexity notions and Jarník sets

Abstract

Given an $\alpha > 1$ and a $\theta$ with unbounded continued fraction entries, we characterise new relations between Sturmian subshifts with slope $\theta$ with respect to (i) an $\alpha$-H\"oder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\theta$, and (iii) complexity notions which we call $\alpha$-repetitive, $\alpha$-repulsive and $\alpha$-finite; generalisations of the properties known as linearly repetitive, repulsive and power free, respectively. We show that the level sets relate naturally to (exact) Jarn\'{\i}k sets and prove that their Hausdorff dimension is $2/(\alpha + 1)$.