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 λ and the prefixes of a right-infinite word ρ ρ . For a positive integer n , let g ( n ) be n minus the maximum length of overlaps between the suffix of λ λ and the prefix of ρ ρ of length n . In a recent publication we have shown that the function g has finite image if and only if λ λ and ρ ρ are ultimately periodic words with a same root. In this paper we give an asymptotic characterization of words λ λ and ρ 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 ( g ( n ) / n ) n tends to zero