Repetition Avoidance in Circular Factors

TL;DR

This paper explores a new variation of repetition avoidance in words, focusing on circular factors, and determines optimal avoidance exponents for small alphabets with bounds for larger ones.

Contribution

It introduces a novel circular factor avoidance problem and establishes the best avoidance exponents for small alphabets, providing bounds for larger ones.

Findings

Optimal avoidance exponent for alphabet size 2

Optimal avoidance exponent for alphabet size 3

Lower bounds for larger alphabets

Abstract

We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a "circular word", i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and provide a lower bound for larger alphabets.