Asymptotic repetitive threshold of balanced sequences

TL;DR

This paper investigates the asymptotic repetition thresholds of balanced sequences over finite alphabets, showing that for any alphabet size greater than one, sequences can have arbitrarily low asymptotic repetition, and identifies minimal thresholds for balanced sequences.

Contribution

It introduces the concept of asymptotic critical exponent for sequences and determines minimal thresholds for balanced sequences over small alphabets.

Findings

Existence of sequences with arbitrarily close to 1 asymptotic critical exponent for any alphabet size > 1.

Balanced sequences have a strictly higher lower bound on their asymptotic critical exponent.

Method to find balanced sequences with minimal asymptotic critical exponent for 2 to 10 symbols.

Abstract

The critical exponent $E(\mathbf u)$ of an infinite sequence $\mathbf u$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf u$. By the famous Dejean's theorem, $E(\mathbf u) \geq 1+\frac1{d-1}$ for every $d$-ary sequence $\mathbf u$. We define the asymptotic critical exponent $E^*(\mathbf u)$ as the upper limit of the maximal repetition of factors of length $n$. We show that for any $d>1$ there exists a $d$-ary sequence $\mathbf u$ having $E^*(\mathbf u)$ arbitrarily close to $1$. Then we focus on the class of $d$-ary balanced sequences. In this class, the values $E^*(\mathbf u)$ are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a $d$-ary balanced sequence with the least asymptotic critical exponent for $2\leq d\leq 10$.