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 three 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.