On some periodic continued fractions along the $\mathbb{Z}_2$ extension over $\mathbb{Q}$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912324033445888 |
|---|---|
| author | Kanamura, Yoshinori Yoshizaki, Hyuga |
| author_facet | Kanamura, Yoshinori Yoshizaki, Hyuga |
| contents | In 2021, Brock, Elkies, and Jordan generalized the theory of periodic continued fractions (PCFs) over $\mathbb{Z}$ to the ring of integers in a number field. In particular, they considered the case where the number field is an intermediate field of the $\mathbb{Z}_2$-extension over $\mathbb{Q}$ and asked whether a $(N, \ell)$-type PCF for $X_n = 2\cos(2π/2^{n+2})$ exists. In this paper, we construct $(1,2)$ and $(0,3)$-type PCFs for $X_n$ for all $n\geq1$. To the best of our knowledge, this is the first explicit construction of type (0,3) continued fractions for all $n\geq1$. To obtain such results, for each type, we construct a bijection between a certain subset of the group of relative units in each layer of the $\mathbb{Z}_2$-extension and the set of PCFs for $X_n$. While our result confirms the existence of such PCFs for all $n\geq1$ in types $(1,2)$ and $(0,3)$, determining all PCFs remains an open problem. The bijections constructed in our result translate this problem into the study of the subsets of the relative units. As a second main result, we give explicit bounds for the logarithms of the relative units corresponding to $(1,2)$ or $(0,3)$-type PCFs for $X_n$. These bounds allow us to explain interesting phenomena observed in the distribution of such points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some periodic continued fractions along the $\mathbb{Z}_2$ extension over $\mathbb{Q}$ Kanamura, Yoshinori Yoshizaki, Hyuga Number Theory Primary 11J70, Secondary 11A55, 11R18, 11R27 In 2021, Brock, Elkies, and Jordan generalized the theory of periodic continued fractions (PCFs) over $\mathbb{Z}$ to the ring of integers in a number field. In particular, they considered the case where the number field is an intermediate field of the $\mathbb{Z}_2$-extension over $\mathbb{Q}$ and asked whether a $(N, \ell)$-type PCF for $X_n = 2\cos(2π/2^{n+2})$ exists. In this paper, we construct $(1,2)$ and $(0,3)$-type PCFs for $X_n$ for all $n\geq1$. To the best of our knowledge, this is the first explicit construction of type (0,3) continued fractions for all $n\geq1$. To obtain such results, for each type, we construct a bijection between a certain subset of the group of relative units in each layer of the $\mathbb{Z}_2$-extension and the set of PCFs for $X_n$. While our result confirms the existence of such PCFs for all $n\geq1$ in types $(1,2)$ and $(0,3)$, determining all PCFs remains an open problem. The bijections constructed in our result translate this problem into the study of the subsets of the relative units. As a second main result, we give explicit bounds for the logarithms of the relative units corresponding to $(1,2)$ or $(0,3)$-type PCFs for $X_n$. These bounds allow us to explain interesting phenomena observed in the distribution of such points. |
| title | On some periodic continued fractions along the $\mathbb{Z}_2$ extension over $\mathbb{Q}$ |
| topic | Number Theory Primary 11J70, Secondary 11A55, 11R18, 11R27 |
| url | https://arxiv.org/abs/2503.09909 |