The repetition threshold for words on $n$ letters, denoted $\mbox{ RT}(n)$, 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 $\mbox{ CRT}_{ \mbox{ W}}(n)$, $\mbox{ CRT}_{ \mbox{ I}}(n)$, and $\mbox{ CRT}_{ \mbox{ S}}(n)$, respectively. Currie and the present authors conjectured that $\mbox{ CRT}_{ \mbox{ I}}(n)=\mbox{ CRT}_{ \mbox{ W}}(n)=\mbox{ RT}(n)$ for all $n\geq 4$. We prove that $\mbox{ CRT}_{ \mbox{ W}}(n)=\mbox{ RT}(n)$ for all $n\geq 45$, which confirms a weak version of this conjecture for all but finitely many values of $n$.
Categories: math.CO cs.DM cs.FL