{"paper":{"title":"Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\\mathbb Q[x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Ester Sgallov\\'a, Jana Glivick\\'a, Jan \\v{S}aroch","submitted_at":"2022-04-14T10:38:31Z","abstract_excerpt":"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 simul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.06866","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2204.06866/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}