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

We show how to construct discretely ordered principal ideal subrings of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q left-bracket x right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}[x]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with various types of prime behaviour. Given any set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of finite strictly increasing sequences <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d 1 comma d 2 comma ellipsis comma d Subscript l Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d_1,d_2,\dots , d_l)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of positive integers such that, for each prime integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet p double-struck upper Z comma d 1 plus p double-struck upper Z comma ellipsis comma d Subscript l Baseline plus p double-struck upper Z EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{p\mathbb {Z}, d_1+p\mathbb {Z},\dots , d_l+p\mathbb {Z}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> does not contain all the cosets modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we can stipulate to have, for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d 1 comma ellipsis comma d Subscript l Baseline right-parenthesis element-of script upper D"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">(d_1,\dots , d_l)\in \mathcal D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a cofinal set of progressions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma f plus d 1 comma ellipsis comma f plus d Subscript l Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(f, f+d_1, \dots , f+d_l)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of prime elements in our principal ideal domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript tau"> <mml:semantics> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>τ<!-- τ --></mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">R_\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, we can simultaneously guarantee that each positive prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g element-of upper R Subscript tau minus double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>τ<!-- τ --></mml:mi> </mml:msub> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">g\in R_\tau \setminus \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either in a prescribed progression as above or there is no other prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript tau"> <mml:semantics> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>τ<!-- τ --></mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">R_\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g minus h element-of double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>h</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">g-h\in \mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper Z With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat {\mathbb {Z}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of profinite integers.