# Two strings at Hamming distance 1 cannot be both quasiperiodic

## TL;DR

This paper extends a known property of periodic strings to quasiperiodic strings, proving that two strings differing at one position cannot both be quasiperiodic, and offers new insights into quasiperiodic combinatorics.

## Contribution

It generalizes a classical fact from periodicity to quasiperiodicity, providing a new theoretical result and insights in combinatorics on words.

## Key findings

- Two strings differing at one position cannot both be quasiperiodic
- New theoretical insights into quasiperiodic structures
- Extension of known periodicity properties to quasiperiodic strings

## Abstract

We present a generalization of a known fact from combinatorics on words related to periodicity into quasiperiodicity. A string is called periodic if it has a period which is at most half of its length. A string $w$ is called quasiperiodic if it has a non-trivial cover, that is, there exists a string $c$ that is shorter than $w$ and such that every position in $w$ is inside one of the occurrences of $c$ in $w$. It is a folklore fact that two strings that differ at exactly one position cannot be both periodic. Here we prove a more general fact that two strings that differ at exactly one position cannot be both quasiperiodic. Along the way we obtain new insights into combinatorics of quasiperiodicities.