On the Continued Fraction Expansion of Almost All Real Numbers
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914135978016768 |
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| author | Jin, Alex Singh, Shreyas Zhang, Zhuo Hildebrand, AJ |
| author_facet | Jin, Alex Singh, Shreyas Zhang, Zhuo Hildebrand, AJ |
| contents | By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$.
We generalize this result in two directions: First, for certain sets $A\subset\mathbb{N}$, we establish simple explicit formulas for the frequency with which the continued fraction expansion of a random real number contains a digit from the set $A$. For example, we show that digits of the form $p-1$, where $p$ is prime, appear with frequency $\log_2(π^2/6)$.
Second, we obtain a simple formula for the frequency with which a string of $k$ consecutive digits $a$ appears in the continued fraction expansion of a random real number. In particular, when $a=1$, this frequency is given by $|\log_2(1+(-1)^k/F_{k+2})|$, where $F_n$ is the $n$th Fibonacci number.
Finally, we compare the frequencies predicted by these results with actual frequencies found among the first 300 million continued fraction digits of $π$, and we provide strong statistical evidence that the continued fraction expansion of $π$ behaves like that of a random real number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Continued Fraction Expansion of Almost All Real Numbers Jin, Alex Singh, Shreyas Zhang, Zhuo Hildebrand, AJ Number Theory 11K50, 11A55 By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions: First, for certain sets $A\subset\mathbb{N}$, we establish simple explicit formulas for the frequency with which the continued fraction expansion of a random real number contains a digit from the set $A$. For example, we show that digits of the form $p-1$, where $p$ is prime, appear with frequency $\log_2(π^2/6)$. Second, we obtain a simple formula for the frequency with which a string of $k$ consecutive digits $a$ appears in the continued fraction expansion of a random real number. In particular, when $a=1$, this frequency is given by $|\log_2(1+(-1)^k/F_{k+2})|$, where $F_n$ is the $n$th Fibonacci number. Finally, we compare the frequencies predicted by these results with actual frequencies found among the first 300 million continued fraction digits of $π$, and we provide strong statistical evidence that the continued fraction expansion of $π$ behaves like that of a random real number. |
| title | On the Continued Fraction Expansion of Almost All Real Numbers |
| topic | Number Theory 11K50, 11A55 |
| url | https://arxiv.org/abs/2403.16761 |