The $\theta$-adics

Abstract

This paper introduces an archimedean, locally Cantor multi-field $\mathcal{O}_{\theta}$ 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 $1<\theta\in \mathcal{O}_{K}^{\times}$, which plays the same role as the prime $p$ does in $\mathbb{Z}_{p}$; the elements of $\mathcal{O}_{\theta}$ are then greedy Laurent series in the base $\theta$. There is a canonical inclusion of the integers $\mathcal{O}_{K}$ with dense image in $\mathcal{O}_{\theta}$ and the operations of sum and product extend to multi-valued operations having at most three values, making $\mathcal{O}_{\theta}$ 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 $\mathcal{O}_{\theta}$. 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 $\mathcal{O}_{\theta}$, with an eye toward obtaining a finer version of class field theory incorporating the ideal arithmetic of quasicrystal rings.