Abelian Repetitions in Sturmian Words

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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 ≥√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 Fⱼ, j>1, has length Fⱼ(Fⱼ₊₁+Fⱼ₋₁ +1)-2 if j is even or Fⱼ(Fⱼ₊₁+Fⱼ₋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 fⱼ has abelian period equal to Fₙ, where n = j/2 if j = 0, 1, 24, or n = 1 + j/2 if j = 34.