{"paper":{"title":"On denominators of consecutive $\\operatorname{SL}(2,{\\mathbb N})$-saturated Farey fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandru Zaharescu, Cristian Cobeli, Florin P. Boca, Jack Anderson","submitted_at":"2025-08-11T13:07:13Z","abstract_excerpt":"The sequence $({\\mathscr S}_Q)_Q$ of $\\operatorname{SL}(2,{\\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\\mathscr S}_Q := \\{ a/q \\in {\\mathbb Q} \\cap (0,1]: q+a+\\bar{a} \\le Q\\}$, where $\\bar{a}$ is the multiplicative inverse of $a\\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\\mathscr S}_Q$ is dense in the region ${\\mathcal V}:=\\{ (x,y)\\in [0,1]^2 : \\max \\{ (1-3x)/2,2x-1\\} \\le y \\le \\max \\{ x,1-x\\} \\}$, and provide a formula for their distribution in ${\\mathcal V}$ as $Q\\rightarrow \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07951","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/2508.07951/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"}