The Weak Circular Repetition Threshold Over Large Alphabets
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The repetition threshold for words on n letters, denoted (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 _(n), _(n), and _(n), respectively. Currie and the present authors conjectured that _(n)=_(n)=(n) for all n≥ 4. We prove that _(n)=(n) for all n≥ 45, which confirms a weak version of this conjecture for all but finitely many values of n.
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