Circular repetition thresholds on some small alphabets: Last cases of Gorbunova's conjecture

Abstract

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 ${\mathrm{CRT}}_{\mathrm{S}}(4)=\frac{3}{2}$ and ${\mathrm{CRT}}_{\mathrm{S}}(5)=\frac{4}{3}$, confirming a conjecture of Gorbunova and providing the last unknown values of the strong circular repetition threshold. We also prove that ${\mathrm{CRT}}_{\mathrm{I}}(3)={\mathrm{CRT}}_{\mathrm{W}}(3)=\mathrm{RT}(3)=\frac{7}{4}$.

27 pages (including a 6 page appendix). As promised in an earlier version, we have added a proof that the strong circular repetition threshold for five letters is 4/3, completing the last case of Gorbunova's conjecture