Asymptotic behavior of the overlap gap between infinite words.

We proceed with the study of ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word λ and the prefixes of a right-infinite word ρ . For a positive integer n, let g(n) be n minus the maximum length of overlaps between the suffix of λ and the prefix of ρ of length n. In a recent publication we have shown that the function g has finite image if and only if λ and ρ are ultimately periodic words with a same root. In this paper we give an asymptotic characterization of words λ and ρ for which the function g has finite image. We prove that this condition is true if and only if the sequence (g (n) / n) n tends to zero [ABSTRACT FROM AUTHOR]