A word is called $\beta$-free if it has no factors of exponent greater than or equal to $\beta$. The repetition threshold $\mathrm{ RT}(k)$ is the infimum of the set of all $\beta$ such that there are arbitrarily long $k$-ary $\beta$-free words (or equivalently, there are $k$-ary $\beta$-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all $\beta$ such that - there are arbitrarily long $k$-ary $\beta$-free circular words is called the weak circular repetition threshold, denoted $\mathrm{ CRT}_{ \mathrm{ W}}(k)$; - there are $k$-ary $\beta$-free circular words of every sufficiently large length is called the intermediate circular repetition threshold, denoted $\mathrm{ CRT}_{ \mathrm{ I}}(k)$; - there are $k$-ary $\beta$-free circular words of every length is called the strong circular repetition threshold, denoted $\mathrm{ CRT}_{ \mathrm{ S}}(k)$. We prove that $\mathrm{ CRT}_{ \mathrm{ S}}(4)=\tfrac{ 3}{ 2}$ and $\mathrm{ CRT}_{ \mathrm{ S}}(5)=\tfrac{ 4}{ 3}$, confirming a conjecture of Gorbunova and providing the last unknown values of the strong circular repetition threshold. We also prove that $\mathrm{ CRT}_{ \mathrm{ I}}(3)=\mathrm{ CRT}_{ \mathrm{ W}}(3)=\mathrm{ RT}(3)=\tfrac{ 7}{ 4}$.
Categories: math.CO cs.FL