{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,11]],"date-time":"2026-09-11T18:33:23Z","timestamp":1789151603328,"version":"build-2803163510"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T00:00:00Z","timestamp":1653004800000},"content-version":"unspecified","delay-in-days":139,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Forum of Mathematics, Pi"],"published-print":{"date-parts":[[2022]]},"abstract":"<jats:title>Abstract<\/jats:title>\n\t  <jats:p>Define the <jats:italic>Collatz map<\/jats:italic><jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline1.png\"\/>\n\t\t<jats:tex-math>\n${\\operatorname {Col}} \\colon \\mathbb {N}+1 \\to \\mathbb {N}+1$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> on the positive integers <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline2.png\"\/>\n\t\t<jats:tex-math>\n$\\mathbb {N}+1 = \\{1,2,3,\\dots \\}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> by setting <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline3.png\"\/>\n\t\t<jats:tex-math>\n${\\operatorname {Col}}(N)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> equal to <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline4.png\"\/>\n\t\t<jats:tex-math>\n$3N+1$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> when <jats:italic>N<\/jats:italic> is odd and <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline5.png\"\/>\n\t\t<jats:tex-math>\n$N\/2$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> when <jats:italic>N<\/jats:italic> is even, and let <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline6.png\"\/>\n\t\t<jats:tex-math>\n${\\operatorname {Col}}_{\\min }(N) := \\inf _{n \\in \\mathbb {N}} {\\operatorname {Col}}^n(N)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> denote the minimal element of the Collatz orbit <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline7.png\"\/>\n\t\t<jats:tex-math>\n$N, {\\operatorname {Col}}(N), {\\operatorname {Col}}^2(N), \\dots $\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>. The infamous <jats:italic>Collatz conjecture<\/jats:italic> asserts that <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline8.png\"\/>\n\t\t<jats:tex-math>\n${\\operatorname {Col}}_{\\min }(N)=1$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> for all <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline9.png\"\/>\n\t\t<jats:tex-math>\n$N \\in \\mathbb {N}+1$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>. Previously, it was shown by Korec that for any <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline10.png\"\/>\n\t\t<jats:tex-math>\n$\\theta&gt; \\frac {\\log 3}{\\log 4} \\approx 0.7924$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>, one has <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline11.png\"\/>\n\t\t<jats:tex-math>\n${\\operatorname {Col}}_{\\min }(N) \\leq N^\\theta $\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> for almost all <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline12.png\"\/>\n\t\t<jats:tex-math>\n$N \\in \\mathbb {N}+1$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> (in the sense of natural density). In this paper, we show that for <jats:italic>any<\/jats:italic> function <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline13.png\"\/>\n\t\t<jats:tex-math>\n$f \\colon \\mathbb {N}+1 \\to \\mathbb {R}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> with <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline14.png\"\/>\n\t\t<jats:tex-math>\n$\\lim _{N \\to \\infty } f(N)=+\\infty $\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>, one has <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline15.png\"\/>\n\t\t<jats:tex-math>\n${\\operatorname {Col}}_{\\min }(N) \\leq f(N)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> for almost all <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline16.png\"\/>\n\t\t<jats:tex-math>\n$N \\in \\mathbb {N}+1$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline17.png\"\/>\n\t\t<jats:tex-math>\n$3$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-adic cyclic group <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000087_inline18.png\"\/>\n\t\t<jats:tex-math>\n$\\mathbb {Z}\/3^n\\mathbb {Z}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.<\/jats:p>","DOI":"10.1017\/fmp.2022.8","type":"journal-article","created":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T11:13:21Z","timestamp":1653045201000},"update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":29,"title":["Almost all orbits of the Collatz map attain almost bounded values"],"prefix":"10.1017","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0140-7641","authenticated-orcid":false,"given":"Terence","family":"Tao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2022,5,20]]},"reference":[{"key":"S2050508622000087_r8","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(77)90087-1"},{"key":"S2050508622000087_r6","unstructured":"[6] Chamberland, M. , A $3x+1$ survey: number theory and dynamical systems, The ultimate challenge: the $3x+1$ problem, 57\u201378, Amer. 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Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"e12"}}