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