Controlling distribution of prime sequences in discretely ordered principal ideal subrings of ℚ[𝕩]

Abstract

We show how to construct discretely ordered principal ideal subrings of Q [ x ] \mathbb {Q}[x] with various types of prime behaviour. Given any set D \mathcal D consisting of finite strictly increasing sequences ( d 1 , d 2 , … , d l ) (d_1,d_2,\dots , d_l) of positive integers such that, for each prime integer p p , the set { p Z , d 1 + p Z , … , d l + p Z } \{p\mathbb {Z}, d_1+p\mathbb {Z},\dots , d_l+p\mathbb {Z}\} does not contain all the cosets modulo p p , we can stipulate to have, for each ( d 1 , … , d l ) ∈ D (d_1,\dots , d_l)\in \mathcal D , a cofinal set of progressions ( f , f + d 1 , … , f + d l ) (f, f+d_1, \dots , f+d_l) of prime elements in our principal ideal domain R τ R_\tau . Moreover, we can simultaneously guarantee that each positive prime g ∈ R τ ∖ N g\in R_\tau \setminus \mathbb {N} is either in a prescribed progression as above or there is no other prime h h in R τ R_\tau such that g − h ∈ Z g-h\in \mathbb {Z} . Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring Z ^ \widehat {\mathbb {Z}} of profinite integers.