The Weak Circular Repetition Threshold Over Large Alphabets

The repetition threshold for words on n letters, denoted RT(n), is the infimum of the set of all r such that there are arbitrarily long r-free words over n letters. A repetition threshold for circular words on n letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for n letters, denoted CRTW(n), CRTI(n), and CRTS(n), respectively. Currie and the present authors conjectured that CRTI(n) = CRTW(n) = RT(n) for all n ≥ 4. We prove that CRTW(n) = RT(n) for all n ≥ 45, which confirms a weak version of this conjecture for all but finitely many values of n.