Abelian Repetitions in Sturmian Words

TL;DR

This paper studies abelian repetitions in Sturmian words, especially Fibonacci words, providing formulas for their maximal exponents and minimal abelian periods using number theory and combinatorics.

Contribution

It introduces a novel analysis of abelian repetitions in Sturmian words, deriving exact formulas for Fibonacci words' abelian periods and maximal exponents.

Findings

The ratio of maximal abelian repetition exponent to period is at least √5 in Sturmian words.

Exact lengths of longest abelian repetitions in Fibonacci words are established.

Fibonacci words have specific abelian periods depending on their index.

Abstract

We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of elementary Number Theory. We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality holds for the Fibonacci infinite word. We further prove that the longest prefix of the Fibonacci infinite word that is an abelian repetition of period $F_j$, $j>1$, has length $F_j(F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or $F_j(F_{j+1}+F_{j-1})-2$ if $j$ is odd. This allows us to give an exact formula for the smallest abelian periods of the Fibonacci finite words. More precisely, we prove that for $j\geq 3$, the Fibonacci word $f_j$ has abelian period equal to $F_n$, where $n = \lfloor{j/2}\rfloor$ if $j = 0, 1, 2\mod{4}$, or $n = 1 + \lfloor{j/2}\rfloor$ if $ j = 3\mod{4}$.