Asymptotic Behaviour of the Maximal Number of Squares in Standard Sturmian Words

Abstract

Denote by sq(w) the number of distinct squares in a string w and let S be the class of standard Sturmian words. They are generalizations of Fibonacci words and are important in combinatorics on words. For Fibonacci words the asymptotic behaviour of the number of runs and the number of squares is the same. We show that for Sturmian words the situation is quite different. The tight bound 8 10 |w| for the number of runs was given in [3]. In this paper we show that the tight bound for the maximal number of squares is 9 10 |w|. We use the results of [11], where exact (but not closed) complicated formulas were given for sq(w) for w ∈ S. We show that for all w ∈ S we have sq(w) ≤ 9 10 |w| and there is an infinite sequence of words w k ∈ S such that lim k→∞ |w k | = ∞ and lim k→∞ sq(w k ) |w k | = 9 10 . Surprisingly the maximal number of squares is reached by the words with recurrences of length only 5. This contrasts with the situation of Fibonacci words, though standard Sturmian words are natural extension of Fibonacci words. If this length drops to 4, the asymptotic behaviour of the maximal number of squares falls down significantly below 9 10 |w|. The structure of Sturmian words rich in squares has been discovered by us experimentally and verified theoretically. The upper bound is much harder, its proof is not a matter of simple calculations. The summation formulas for the number of squares are complicated, no closed formula is known. Some nontrivial reductions were necessary.

Computing the number of cubic runs in standard Sturmian words

“…For the class S of standard Sturmian words, there are known exact formulas for the number of runs and squares and their asymptotic behavior; see [2,22] for details. In this case, we have lim n→∞ ρ(n)…”

The Maximal Number of Runs in Standard Sturmian Words

“…The words of the type b can be considered similarly and all the results hold. For more interesting facts related to combinatorial structure of standard words see for instance in [1], [2], [13], [14] and [15].…”