부산대학교 도서관

Let $\Sigma$ be the binary alphabet $\{0,1\}$ and let $w$ denote a wordover $\Sigma$. An infinite Sturmian word on $\Sigma$ is a word withexactly $n+1$ distinct factors of length $n$ for any $n\geq 0$. It iswell known that a Sturmian word can be given in terms of a slope andits continued fraction expansion $[a_0,a_1,a_2,\dots]$, as recalled inSection 1 of the present paper.\par In [Trans. Amer. Math.Soc. {\bf 371} (2019), no.~5, 3281--3308; MR3896112], Y. Bugeaudand D.~H. Kim introduced the exponent of repetition ${\rm rep}(n,w)$, which is theminimum length of a prefix of $w$, whose prefix of length $n$ isrepeated. This number turns out to be well defined for all $n>0$ in anySturmian word. As an example, let $w$ be the word 0100010, then theprefix of length 2 is repeated twice in the prefix of length~6, so${\rm rep}(2,w)=6$.\par It is known that $\limsup_{n\rightarrow \infty}{\rm rep}(n,w)/n=2$ forany Sturmian word $w$. Let ${\rm rep}(w)$ denote $\liminf_{n\rightarrow\infty}{\rm rep}(n,w)$, so ${\rm rep}(w)\geq 1$. In the aforementionedpaper it was proved that ${\rm rep}(w)\leqr_{\max}=\sqrt{10}-3/2=1.66227\dots$, and equality holds for a certainword $w$, 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 pointin the set of all possible values ${\rm rep}(w)$, for $w$ a Sturmianword.\par In the present paper, the authors prove that the maximum accumulationpoint 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)