Every Nonnegative Real Number Is an Abelian Critical Exponent

Abstract

The abelian critical exponent of an infinite word w is defined as the maximum ratio between the exponent and the period of an abelian power occurring in w. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number θ there exists an infinite word having abelian critical exponent θ. We also extend this result to the k-abelian setting.

On the 2-binomial complexity of the generalized Thue-Morse words

“…Two words u, v ∈ A * are abelian equivalent, denoted by u ∼ ab v, if |u| a = |v| a for every a ∈ A where |u| a denotes the number of occurrences of the letter a in u. The notion has been studied in the relation of abelian complexity of infinite words [4,14,15,23], abelian repetitions and avoidance [3,10,18], and other topics [6,17,21]; see also [20] and references therein.…”

Abelian Closures of Infinite Binary Words

“…Two finite words u and v are called abelian equivalent if, for each letter a of the underlying alphabet Σ, the words u and v contain equally many occurrences of a. The notion has been studied in the relation of abelian complexity of infinite words [2,18,27,28], abelian repetitions and avoidance [3,4,14,23,26], other topics [9,21,22,25]; see also [24] and references therein.…”