On Balanced Sequences and Their Asymptotic Critical Exponent

Abstract

We study aperiodic balanced sequences over finite alphabets. A sequence \(\mathbf {v}\) of this type is fully characterised by a Sturmian sequence \(\mathbf {u}\) and two constant gap sequences \(\mathbf {y}\) and \(\mathbf {y}'\). We study the language of \(\mathbf {v}\), with focus on return words to its factors. We provide a uniform lower bound on the asymptotic critical exponent of all sequences \(\mathbf {v}\) arising by \(\mathbf {y}\) and \(\mathbf {y}'\). It is a counterpart to the upper bound on the least critical exponent of \(\mathbf {v}\) conjectured and partially proved recently in works of Baranwal, Rampersad, Shallit and Vandomme. We deduce a method computing the exact value of the asymptotic critical exponent of \(\mathbf {v}\) provided the associated Sturmian sequence \(\mathbf {u}\) has a quadratic slope. The method is used to compare the critical and the asymptotic critical exponent of balanced sequences over an alphabet of size \(d\le 10\) which are conjectured by Rampersad et al. to have the least critical exponent.