Avoiding approximate repetitions with respect to the longest common subsequence distance

TL;DR

This paper demonstrates the existence of infinite words that avoid approximate repetitions based on the longest common subsequence measure, using entropy compression techniques rooted in the Lovasz local lemma.

Contribution

It introduces a new approach to avoid approximate repetitions in words by employing the longest common subsequence measure and entropy compression methods.

Findings

Existence of words avoiding approximate repetitions with LCS measure

Application of entropy compression method to combinatorial word problems

Extension of previous work using Hamming and edit distances

Abstract

Ochem, Rampersad, and Shallit gave various examples of infinite words avoiding what they called approximate repetitions. An approximate repetition is a factor of the form xx', where x and x' are close to being identical. In their work, they measured the similarity of x and x' using either the Hamming distance or the edit distance. In this paper, we show the existence of words avoiding approximate repetitions, where the measure of similarity between adjacent factors is based on the length of the longest common subsequence. Our principal technique is the so-called "entropy compression" method, which has its origins in Moser and Tardos's algorithmic version of the Lovasz local lemma.