{"paper":{"title":"An approximation of the Collatz map and a lower bound for the average total stopping time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT","math.PR"],"primary_cat":"math.DS","authors_text":"Manuel Inselmann","submitted_at":"2024-02-05T18:34:02Z","abstract_excerpt":"Define the map $\\mathsf{T}$ on the positive integers by $\\mathsf{T}(m)=\\frac{m}{2}$ if $m$ is even and by $\\mathsf{T}(m)=\\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\\epsilon>0$, almost all $m\\in\\mathbb{Z}^+$ (in the sense of natural density) fulfill $(\\frac{\\sqrt{3}}{2})^km^{1-\\epsilon}\\leq \\mathsf{T}^k(m)\\leq (\\frac{\\sqrt{3}}{2})^km^{1+\\epsilon}$ simultaneously for all $0\\leq k\\leq \\alpha\\log m$ with $\\alpha=(\\log 2)^{-1}\\approx 1.443$. We extend this result to $\\alpha=2(\\log\\frac{4}{3})^{-1}\\approx 6.952$, which is the maximally possible value. Set $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.03276","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/2402.03276/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"}