Asymptotic behavior of the overlap gap between infinite words

Abstract

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