Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb{Q}[x]$

Abstract

We show how to construct discretely ordered principal ideal subrings of $\mathbb{Q}[x]$ with various types of prime behaviour. Given any set $\mathcal{D}$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots ,d_l)$ of positive integers such that, for each prime integer $p$, the set $\{p\mathbb{Z},d_1+p\mathbb{Z},\dots ,d_l+p\mathbb{Z}\}$ does not contain all the cosets modulo $p$, we can stipulate to have, for each $(d_1,\dots ,d_l)\in \mathcal{D}$, a cofinal set of progressions $(f,f+d_1,\dots ,f+d_l)$ of prime elements in our principal ideal domain $R_{\tau }$. Moreover, we can simultaneously guarantee that each positive prime $g\in R_{\tau }\setminus \mathbb{N}$ is either in a prescribed progression as above or there is no other prime $h$ in $R_{\tau }$ such that $g-h\in \mathbb{Z}$. Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring $\hat{\mathbb{Z}}$ of profinite integers.

9 pages; mostly minor corrections; some slightly more detailed explanations