Abelian Repetitions in Sturmian Words

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. If k _m denotes the maximal exponent of an abelian repetition of period m, we prove that limsup $k_{m}/m\ge \sqrt{5}$ for any Sturmian word, 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 ≥ 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 $j = 3\mod{4}$.