Repetition threshold for circular words

Abstract

We find the threshold between avoidable and unavoidable repetitions in circular words over k letters for any k ≥ 6. Namely, we show that the number CRT(k), satisfies the following properties. For any n there exists a k-ary circular word of length n containing no repetition of exponent greater than CRT(k). On the other hand, k-ary circular words of some lengths must have a repetition of exponent at least CRT(k).