Repetition Threshold for Circular Words

Abstract

We find the threshold between avoidable and unavoidable repetitions in circular words over $k$ letters for any $k\ge6$. Namely, we show that the number $CRT(k)=\frac{\left\lceil {k/2}\right\rceil{+}1}{\left\lceil {k/2}\right\rceil}$ 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 $CRT(k)$.