On some periodic continued fractions along the $\mathbb{Z}_2$ extension over $\mathbb{Q}$

TL;DR

This work advances the theory of periodic continued fractions in the $\mathbb{Z}_2$-extension by constructing explicit $\mathbb{Z}[X_{n-1}]$-PCFs of types $(1,2)$ and $(0,3)$ for all $X_n$ with $n\ge 2$. It achieves this via bijections between carefully chosen relative units in $\mathbb{Z}[X_n]$ and the PCFs, yielding concrete expansions and illuminating bounds on the growth of conjugates that explain observed distribution patterns. The authors provide two explicit PCF realizations for each $n$, establish a lattice-theoretic framework to constrain all possible PCFs, and connect these PCFs to generalized Pell equations. They also relate the unit structures to class-number phenomena in the $\mathbb{Z}_2$-extension, under plausible conjectures, and highlight open problems in fully classifying all PCFs for these extensions.

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 $X_n = 2\cos(2π/2^{n+2})$ 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.

Figures (2)

• Figure 1: $l_2(RE_2^+(1,2))$
• Figure 2: $l_2(RE_2^+(1,2))$

Theorems & Definitions (24)

• Theorem 1.2
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• Definition 2.4
• Theorem 3.1
• proof