On $k$-abelian Equivalence and Generalized Lagrange Spectra

TL;DR

This paper explores generalized Lagrange spectra derived from $k$-abelian critical exponents of Sturmian words, revealing their density, non-closure, and relation to continued fractions, extending classical spectral results.

Contribution

It characterizes the generalized spectra for $k > 1$ in terms of the classical Lagrange spectrum and analyzes their topological and geometric properties.

Findings

Generalized spectra are dense and non-closed for $k > 1$.

The least accumulation points of these spectra are explicitly described.

The approach links $k$-abelian powers in Sturmian words to continued fractions.

Abstract

We study the set of $k$-abelian critical exponents of all Sturmian words. It has been proven that in the case $k = 1$ this set coincides with the Lagrange spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense non-closed set. This is in contrast with the case $k = 1$, where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra. Our geometric approach allows the study of $k$-abelian powers in Sturmian words by means of continued fractions.