The overlap gap between left-infinite and right-infinite words

TL;DR

This paper characterizes when the set of overlap gaps between left-infinite and right-infinite words is finite, showing it occurs precisely when both words are ultimately periodic with specific repeating structures.

Contribution

It provides a complete characterization of the conditions under which the overlap gap set is finite, linking it to the ultimate periodicity of the infinite words.

Findings

The overlap gap set is finite if and only if both words are ultimately periodic.

Ultimate periodicity involves words of the form $u^{- abla}w_1$ and $w_2 u^{ abla}$.

The result connects the structure of infinite words to the properties of their overlap gaps.

Abstract

Given two finite words $u$ and $v$ of equal length, define the \emph{overlap gap between $u$ and $v$}, denoted $og(u,v)$, as the least integer $m$ for which there exist words $x$ and $x'$ of length $m$ such that $xu=vx'$ or $ux=x'v$. Informally, the overlap gap measures the outside parts of the greatest overlap of the given words. For a left-infinite word $\lambda$ and a right-infinite word $\rho$, let $og_{\lambda,\rho}$ be the function defined, for each non-negative integer $n$, by $og_{\lambda,\rho}(n)=og(\lambda_n,\rho_n)$, where $\lambda_n$ and $\rho_n$ are, respectively, the suffix of $\lambda$ and the prefix of $\rho$ of length $n$. Also, denote by $OG_{\lambda,\rho}$ the image of the function $og_{\lambda,\rho}$. In this paper, we show that $OG_{\lambda,\rho}$ is a finite set if and only if $\lambda$ and $\rho$ are ultimately periodic infinite words of the form $\lambda=u^{-\infty}w_1=\cdots uuuw_1$ and $\rho=w_2u^\infty=w_2uuu\cdots$ for some finite words $u$, $w_1$ and $w_2$.