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A note about words which coincide except in one position

Margot Bruneaux

TL;DR

It is shown that a result about words which coincide except in one position given as an exercise in Lothaire's Algebraic Combinatorics on Words is false and a modified statement is derived which allows to fix the proof of a theorem which originally used the result of this exercise.

Abstract

In this short note, we show that a result about words which coincide except in one position given as an exercise in Lothaire's Algebraic Combinatorics on Words is false. Moreover, we derive a modified statement which allows us to fix the proof of a theorem which originally used the result of this exercise.

A note about words which coincide except in one position

TL;DR

It is shown that a result about words which coincide except in one position given as an exercise in Lothaire's Algebraic Combinatorics on Words is false and a modified statement is derived which allows to fix the proof of a theorem which originally used the result of this exercise.

Abstract

In this short note, we show that a result about words which coincide except in one position given as an exercise in Lothaire's Algebraic Combinatorics on Words is false. Moreover, we derive a modified statement which allows us to fix the proof of a theorem which originally used the result of this exercise.

Paper Structure

This paper contains 3 theorems, 1 equation.

Key Result

Proposition 1

Let $w$ and $v$ be two words having the same length $n$ such that $w$ has period $q$ and $v$ has period $p$. Assume that $w$ and $v$ coincide except, maybe, in one position. If $\max\{p,q\}\leqslant \left\lfloor\frac{n}{2}\right\rfloor$ then $w=v$.

Theorems & Definitions (4)

  • Proposition 1
  • Remark 1
  • Theorem 1
  • Lemma 1