{"paper":{"title":"Almost all orbits of the Collatz map attain almost bounded values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.NT"],"primary_cat":"math.PR","authors_text":"Terence Tao","submitted_at":"2019-09-08T23:15:50Z","abstract_excerpt":"Define the \\emph{Collatz map} $\\mathrm{Col} : \\mathbb{N}+1 \\to \\mathbb{N}+1$ on the positive integers $\\mathbb{N}+1 = \\{1,2,3,\\dots\\}$ by setting $\\mathrm{Col}(N)$ equal to $3N+1$ when $N$ is odd and $N/2$ when $N$ is even, and let $\\mathrm{Col}_{\\min}(N) := \\inf_{n \\in \\mathbb{N}} \\mathrm{Col}^n(N)$ denote the minimal element of the Collatz orbit $N, \\mathrm{Col}(N), \\mathrm{Col}^2(N), \\dots$. The infamous \\emph{Collatz conjecture} asserts that $\\mathrm{Col}_{\\min}(N)=1$ for all $N \\in \\mathbb{N}+1$. Previously, it was shown by Korec that for any $\\theta > \\frac{\\log 3}{\\log 4} \\approx 0.7924"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.03562","kind":"arxiv","version":5},"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/1909.03562/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"}