On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916188062220288 |
|---|---|
| author | Boca, Florin P. Siskaki, Maria |
| author_facet | Boca, Florin P. Siskaki, Maria |
| contents | This note provides an effective bound in the Gauss-Kuzmin-Lévy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $λ(T^{-n}I) =μ(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $λ$ is the Lebesgue measure on $\mathbb R$, $μ$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07452 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions Boca, Florin P. Siskaki, Maria Number Theory Dynamical Systems 11K50, 37A05, 37A44, 47B38 This note provides an effective bound in the Gauss-Kuzmin-Lévy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $λ(T^{-n}I) =μ(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $λ$ is the Lebesgue measure on $\mathbb R$, $μ$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$ |
| title | On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions |
| topic | Number Theory Dynamical Systems 11K50, 37A05, 37A44, 47B38 |
| url | https://arxiv.org/abs/2209.07452 |