Bounds on the closed-rich constant of infinite words

A finite word $w$ is called \textit{closed} if it has length at most 1 or it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences in $w$. An infinite word $u$ is called \textit{closed-rich} if the infimum of all possible ratios between the number of closed factors within any factor $w$ of $u$ and square of the length of $w$ exists and is positive. We define this infimum as the closed-rich constant $C_u$ of the infinite closed-rich word $u$. Puzynina and Parshina (2024) proved that infinite closed-rich words exist. In this paper, we study possible values of closed-rich constants of infinite closed-rich words. In particular, we estimate the supremum $C_{sup}$ of the closed-rich constants of infinite closed-rich words: we show that $C_{sup} \leq 0.165952$. Besides that, we study the closed-rich constant $C_f$ of the Fibonacci word $f$ and show that $0.09519 \leq C_f\leq 0.10893$. In particular, this gives a lower bound for $C_{sup}$: $0.09519 \leq C_{sup}$.