# Every nonnegative real number is an abelian critical exponent

## TL;DR

This paper proves that for every nonnegative real number, there exists an infinite word with an abelian critical exponent equal to that number, extending the result to the $k$-abelian setting.

## Contribution

It constructs infinite words with any prescribed abelian critical exponent, filling all gaps in the spectrum, and extends the result to the $k$-abelian case.

## Key findings

- Every nonnegative real number is an abelian critical exponent of some infinite word.
- The set of abelian critical exponents includes all positive real numbers, filling the spectrum.
- Extension of the result to the $k$-abelian setting.

## Abstract

The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number $\theta$ there exists an infinite word having abelian critical exponent $\theta$. We also extend this result to the $k$-abelian setting.