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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2511.01992 |
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| _version_ | 1866914136794857472 |
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| author | Singh, Shreyas Zhang, Zhuo Hildebrand, AJ |
| author_facet | Singh, Shreyas Zhang, Zhuo Hildebrand, AJ |
| contents | By a classical result of Gauss and Kuzmin, the frequency with which a string $\mathbf{a}=(a_1,\dots,a_n)$ of positive integers appears in the continued fraction expansion of a random real number is given by $μ_{GK}({I(\mathbf{a})})$, where $I(\mathbf{a})$ is the set of real numbers in $[0,1)$ whose continued fraction expansion begins with the string $\mathbf{a}$ and $μ_{GK}$ is the \emph{Gauss--Kuzmin measure}, defined by $μ_{GK}(I)= \frac{1}{\log 2}\int_I \frac{1}{1+x} dx$, for any interval $I\subseteq[0,1]$. % It is known that the Gauss--Kuzmin measure satisfies the symmetry property $(*)$ $μ_{GK}(I(\mathbf{a}))=μ_{GK}(I(\overleftarrow{\mathbf{a}}))$, where $\overleftarrow{\mathbf{a}}=(a_n,\dots,a_1)$ is the reverse of the string $\mathbf{a}$. We show that this property in fact characterizes the Gauss--Kuzmin measure: If $μ$ is any probability measure with continuous density function on $[0,1]$ satisfying $μ(I(\mathbf{a}))=μ(I(\overleftarrow{\mathbf{a}}))$ for all finite strings $\mathbf{a}$, then $μ=μ_{GK}$. % We also consider the question whether symmetries analogous to $(*)$ hold for permutations of $\mathbf{a}$ other than the reverse $\overleftarrow{\mathbf{a}}$; we call such a symmetry \emph{nontrivial}. We show that strings $\mathbf{a}$ of length $3$ have no nontrivial symmetries, while for each $n\ge 4$ there exists an infinite family of strings $\mathbf{a}$ of length $n$ that do have nontrivial symmetries. Finally we present numerical data supporting the conjecture that, in an appropriate asymptotic sense, ``almost all'' strings $\mathbf{a}$ have no nontrivial symmetries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Elementary Characterization of the Gauss--Kuzmin Measure in the Theory of Continued Fractions Singh, Shreyas Zhang, Zhuo Hildebrand, AJ Number Theory 11K50 By a classical result of Gauss and Kuzmin, the frequency with which a string $\mathbf{a}=(a_1,\dots,a_n)$ of positive integers appears in the continued fraction expansion of a random real number is given by $μ_{GK}({I(\mathbf{a})})$, where $I(\mathbf{a})$ is the set of real numbers in $[0,1)$ whose continued fraction expansion begins with the string $\mathbf{a}$ and $μ_{GK}$ is the \emph{Gauss--Kuzmin measure}, defined by $μ_{GK}(I)= \frac{1}{\log 2}\int_I \frac{1}{1+x} dx$, for any interval $I\subseteq[0,1]$. % It is known that the Gauss--Kuzmin measure satisfies the symmetry property $(*)$ $μ_{GK}(I(\mathbf{a}))=μ_{GK}(I(\overleftarrow{\mathbf{a}}))$, where $\overleftarrow{\mathbf{a}}=(a_n,\dots,a_1)$ is the reverse of the string $\mathbf{a}$. We show that this property in fact characterizes the Gauss--Kuzmin measure: If $μ$ is any probability measure with continuous density function on $[0,1]$ satisfying $μ(I(\mathbf{a}))=μ(I(\overleftarrow{\mathbf{a}}))$ for all finite strings $\mathbf{a}$, then $μ=μ_{GK}$. % We also consider the question whether symmetries analogous to $(*)$ hold for permutations of $\mathbf{a}$ other than the reverse $\overleftarrow{\mathbf{a}}$; we call such a symmetry \emph{nontrivial}. We show that strings $\mathbf{a}$ of length $3$ have no nontrivial symmetries, while for each $n\ge 4$ there exists an infinite family of strings $\mathbf{a}$ of length $n$ that do have nontrivial symmetries. Finally we present numerical data supporting the conjecture that, in an appropriate asymptotic sense, ``almost all'' strings $\mathbf{a}$ have no nontrivial symmetries. |
| title | An Elementary Characterization of the Gauss--Kuzmin Measure in the Theory of Continued Fractions |
| topic | Number Theory 11K50 |
| url | https://arxiv.org/abs/2511.01992 |