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On some periodic continued fractions along the $\mathbb{Z}_2$ extension over $\mathbb{Q}$
Number Theory · Submitted 12 Mar 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we construct $(1,2)$ and $(0,3)$-type PCFs for $X_n$ for all $n\geq1$.
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The abstract, sentence by sentence
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 $X_n = 2\cos(2\pi/2^{n+2})$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | 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 $X_n = 2\cos(2\pi/2^{n+2})$ exists. | background 0.96 | 0.04 | 0.03 | asserted 0.98 |
| s2 | In this paper, we construct $(1,2)$ and $(0,3)$-type PCFs for $X_n$ for all $n\geq1$. | result 0.56 | 0.98 | 0.81 | asserted 1.00 |
| s3 | To the best of our knowledge, this is the first explicit construction of type (0,3) continued fractions for all $n\geq1$. | contribution 0.49 | 0.78 | 0.21 | asserted 0.91 |
| s4 | 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 $X_n$. | method 0.99 | 0.96 | 0.07 | asserted 1.00 |
| s5 | 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. | limitation 0.91 | 0.75 | 0.31 | asserted 0.99 |
| s6 | The bijections constructed in our result translate this problem into the study of the subsets of the relative units. | interpretation 0.87 | 0.93 | 0.10 | asserted 1.00 |
| s7 | 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 $X_n$. | result 0.80 | 0.97 | 0.21 | asserted 1.00 |
| s8 | These bounds allow us to explain interesting phenomena observed in the distribution of such points. | interpretation 0.99 | 0.89 | 0.10 | asserted 0.98 |
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The paper
Original abstract
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 $X_n = 2\cos(2\pi/2^{n+2})$ 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.
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