On minimal critical exponent of balanced sequences

TL;DR

This paper investigates the minimal critical exponent of balanced sequences over finite alphabets, refuting a previous conjecture for larger alphabets and proposing a new lower bound for the critical exponent.

Contribution

It disproves the existing conjecture for alphabet sizes $d \\geq 11$ and establishes a new lower bound of \frac{d-1}{d-2} for these cases.

Findings

Critical exponents for $d \geq 11$ are at least \frac{d-1}{d-2}.

The bound is achieved for all even $d \geq 12$.

The conjecture is refuted for larger alphabets.

Abstract

We study the threshold between avoidable and unavoidable repetitions in infinite balanced sequences over finite alphabets. The conjecture stated by Rampersad, Shallit and Vandomme says that the minimal critical exponent of balanced sequences over the alphabet of size $d \geq 5$ equals $\frac{d-2}{d-3}$. This conjecture is known to hold for $d\in \{5, 6, 7,8,9,10\}$. We refute this conjecture by showing that the picture is different for bigger alphabets. We prove that critical exponents of balanced sequences over an alphabet of size $d\geq 11$ are lower bounded by $\frac{d-1}{d-2}$ and this bound is attained for all even numbers $d\geq 12$. According to this result, we conjecture that the least critical exponent of a balanced sequence over $d$ letters is $\frac{d-1}{d-2}$ for all $d\geq 11$.