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.
Categories: math.NT