부산대학교 도서관

Let $\Sigma$ be the binary alphabet $\{0,1\}$ and let $w$ denote a word over $\Sigma$. An infinite Sturmian word on $\Sigma$ is a word with exactly $n+1$ distinct factors of length $n$ for any $n\geq 0$. It is well known that a Sturmian word can be given in terms of a slope and its continued fraction expansion $[a_0,a_1,a_2,\dots]$, as recalled in Section 1 of the present paper. \par In [Trans. Amer. Math. Soc. {\bf 371} (2019), no.~5, 3281--3308; MR3896112], Y. Bugeaud and D.~H. Kim introduced the exponent of repetition ${\rm rep}(n,w)$, which is the minimum length of a prefix of $w$, whose prefix of length $n$ is repeated. This number turns out to be well defined for all $n>0$ in any Sturmian word. As an example, let $w$ be the word 0100010, then the prefix of length 2 is repeated twice in the prefix of length~6, so ${\rm rep}(2,w)=6$. \par It is known that $\limsup_{n\rightarrow \infty}{\rm rep}(n,w)/n=2$ for any Sturmian word $w$. Let ${\rm rep}(w)$ denote $\liminf_{n\rightarrow \infty}{\rm rep}(n,w)$, so ${\rm rep}(w)\geq 1$. In the aforementioned paper it was proved that ${\rm rep}(w)\leq r_{\max}=\sqrt{10}-3/2=1.66227\dots$, and equality holds for a certain word $w$, whose continued fraction expansion has the form $[0,a_1,\dots,a_K,\overline{2,1,1}]$, for suitable positive integers $K$ and $a_i\geq 1$. This value $r_{\max}$ is not an accumulation point in the set of all possible values ${\rm rep}(w)$, for $w$ a Sturmian word. \par In the present paper, the authors prove that the maximum accumulation point of ${\rm rep}(w)$ is $$ r_1=\frac{48+\sqrt{10}}{31}=1.65039\dots, $$ and there is no Sturmian $w$ word with $r_1<{\rm rep}(w)