Bounds on the closed-rich constant of infinite words
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| Format: | Preprint |
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2026
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| _version_ | 1866916027475951616 |
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| author | Maity, Anuran Puzynina, Svetlana |
| author_facet | Maity, Anuran Puzynina, Svetlana |
| contents | 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}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_19535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounds on the closed-rich constant of infinite words Maity, Anuran Puzynina, Svetlana Combinatorics 68R15 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}$. |
| title | Bounds on the closed-rich constant of infinite words |
| topic | Combinatorics 68R15 |
| url | https://arxiv.org/abs/2605.19535 |