Structure and growth of $\mathbb{R}$-bonacci words
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909749958672384 |
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| author | Dovgal, Sergey Kirgizov, Sergey |
| author_facet | Dovgal, Sergey Kirgizov, Sergey |
| contents | A binary word is called $q$-decreasing, for $q>0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. We show that for any real positive $q$ the number of $q$-decreasing words of length $n$ grows as $C_q \cdot Φ(q)^n$ for some constant $C_q$ which depends on $q$ but not on $n$. From previous works, it is already known that $Φ(1)$ is the golden ratio, $Φ(2)$ is equal to the tribonacci constant, $Φ(k)$ is $(k+1)$-bonacci constant. We prove that the function $Φ(q)$ is strictly increasing, discontinuous at every positive rational point, and exhibits a fractal structure related to the Stern-Brocot tree and Minkowski's question mark function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_01213 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structure and growth of $\mathbb{R}$-bonacci words Dovgal, Sergey Kirgizov, Sergey Combinatorics Discrete Mathematics 05A05, 68R15, 11B39 A binary word is called $q$-decreasing, for $q>0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. We show that for any real positive $q$ the number of $q$-decreasing words of length $n$ grows as $C_q \cdot Φ(q)^n$ for some constant $C_q$ which depends on $q$ but not on $n$. From previous works, it is already known that $Φ(1)$ is the golden ratio, $Φ(2)$ is equal to the tribonacci constant, $Φ(k)$ is $(k+1)$-bonacci constant. We prove that the function $Φ(q)$ is strictly increasing, discontinuous at every positive rational point, and exhibits a fractal structure related to the Stern-Brocot tree and Minkowski's question mark function. |
| title | Structure and growth of $\mathbb{R}$-bonacci words |
| topic | Combinatorics Discrete Mathematics 05A05, 68R15, 11B39 |
| url | https://arxiv.org/abs/2310.01213 |