A Generalization of Repetition Threshold

Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number α such that there exists an infinite word over a k-letter alphabet that avoids β-powers for all β > α. We generalize this concept to include the lengths of the avoided words. We give some conjectures supported by numerical evidence and prove some of these conjectures. As a consequence of one of our results, we show that the pattern ABCBABC is 2-avoidable. This resolves a question left open in Cassaigne’s thesis. Key words: Combinatorics on words, Repetitions

• Type: Conference
• Year: 2004
• Where: MFCS
• Authors: Lucian Ilie, Pascal Ochem, Jeffrey Shallit
• Keywords: Avoided Words, Infinite Word, MFCS 2004, Smallest Real Number