The $θ$-adics
T. M. Gendron, A. Zenteno
TL;DR
This work builds an archimedean analog of $p$-adic arithmetic by constructing the $\theta$-adic world for a real quadratic field $K=\mathbb{Q}(\theta)$ with a unit $\theta>1$. It defines Greedy expansions and the $\theta$-integers $\mathbb{Z}_{\theta}$ as a quasicrystal-like model set, then introduces an infranorm $|\cdot|_{\theta}$ whose completion $\widehat{\mathcal{O}}_{\theta}$ is a field isomorphic to $\mathbb{R}$; the original ring $\mathcal{O}_{K}$ densely embeds into a Marty multifield $\mathcal{O}_{\theta}$ with multivalued arithmetic. The paper proves that $\mathcal{O}_{\theta}$ maps onto its completion with fibers of size at most three, and that $\mathcal{O}_{\theta}$ admits a locally Cantor topology where addition and multiplication are multicontinuous. Together, these results yield a robust framework for an arithmetized place at infinity, potentially enabling a quasicrystal-inspired class field theory that coherently includes infinite places alongside the finite ones.
Abstract
This paper introduces an archimedean, locally Cantor multi-field $\mathcal{O}_θ$ which gives an analog of the $p$-adic number field at a place at infinity of a real quadratic extension $K$ of $\mathbb{Q}$. This analog is defined using a unit $1<θ\in \mathcal{O}_{K}^{\times}$, which plays the same role as the prime $p$ does in $\mathbb{Z}_{p}$; the elements of $\mathcal{O}_θ$ are then greedy Laurent series in the base $θ$. There is a canonical inclusion of the integers $\mathcal{O}_{K}$ with dense image in $\mathcal{O}_θ$ and the operations of sum and product extend to multi-valued operations having at most three values, making $\mathcal{O}_θ$ a multi-field in the sense of Marty. We show that the (geometric) completions of 1-dimensional quasicrystals contained in $\mathcal{O}_{K}$ map canonically to $\mathcal{O}_θ$. The motivation for this work arises in part from a desire to obtain a more arithmetic treatment of a place at infinity by replacing $\mathbb{R}$ with $\mathcal{O}_θ$, with an eye toward obtaining a finer version of class field theory incorporating the ideal arithmetic of quasicrystal rings.