On the Asymptotic Palindrome Density of Fibonacci Infinite Words
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| Format: | Preprint |
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2025
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| _version_ | 1866908773949374464 |
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| author | Abdullah, Duaa Hamoud, Jasem |
| author_facet | Abdullah, Duaa Hamoud, Jasem |
| contents | In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(λ,n)$, $\mathrm{dens}(α,n)$, and $\mathrm{dens}(β,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{φ_1}$, where $φ_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Asymptotic Palindrome Density of Fibonacci Infinite Words Abdullah, Duaa Hamoud, Jasem Combinatorics 05C42, 11B05, 11R45, 11B39, 68R15 G.2.0; F.2.2 In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(λ,n)$, $\mathrm{dens}(α,n)$, and $\mathrm{dens}(β,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{φ_1}$, where $φ_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities. |
| title | On the Asymptotic Palindrome Density of Fibonacci Infinite Words |
| topic | Combinatorics 05C42, 11B05, 11R45, 11B39, 68R15 G.2.0; F.2.2 |
| url | https://arxiv.org/abs/2511.06485 |