The repetition threshold of episturmian sequences

TL;DR

This paper determines the exact repetition threshold for the class of d-ary episturmian sequences, showing it is achieved by the d-bonacci sequence and expressed as a specific algebraic number.

Contribution

It establishes the precise repetition threshold for d-ary episturmian sequences, extending the understanding of repetition properties in generalized Sturmian sequence classes.

Findings

Repetition threshold for d-ary episturmian sequences is 2 + 1/(t-1).

Threshold is achieved by the d-bonacci sequence.

The value is the unique positive root of x^d - x^{d-1} - ... - x - 1.

Abstract

The repetition threshold of a class $C$ of infinite $d$-ary sequences is the smallest real number $r$ such that in the class $C$ there exists a sequence that avoids $e$-powers for all $e> r$. This notion was introduced by Dejean in 1972 for the class of all sequences over a $d$-letter alphabet. Thanks to the effort of many authors over more than 30 years, the precise value of the repetition threshold in this class is known for every $d \in \mathbb N$. The repetition threshold for the class of Sturmian sequences was determined by Carpi and de Luca in 2000. Sturmian sequences may be equivalently defined in various ways, therefore there exist many generalizations to larger alphabets. Rampersad, Shallit and Vandome in 2020 initiated a study of the repetition threshold for the class of balanced sequences -- one of the possible generalizations of Sturmian sequences. Here, we focus on the class of $d$-ary episturmian sequences -- another generalization of Sturmian sequences introduced by Droubay, Justin and Pirillo in 2001. We show that the repetition threshold of this class is reached by the $d$-bonacci sequence and its value equals $2+\frac{1}{t-1}$, where $t>1$ is the unique positive root of the polynomial $x^d-x^{d-1}- \cdots -x-1$.