For any function f tending to infinity, almost all N in logarithmic density have Collatz orbit minimum below f(N).
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
This is Terence Tao's theorem that almost all Collatz orbits attain almost bounded values: for any function f with f(N) tending to infinity, however slowly (log log log log N, for instance), the minimum of the Collatz orbit of N is below f(N) for almost all N, where "almost all" is measured in logarithmic density. It was announced on arXiv in September 2019 and published after refereeing in Forum of Mathematics, Pi in 2022. It is the strongest known partial result toward the Collatz conjecture, sharpening Korec's theorem that almost all N descend below N^θ for any θ above log 3/log 4 from power functions to arbitrary functions tending to infinity, at the cost of passing from natural density to the weaker logarithmic density.
The proof stands as an accepted result: it has been refereed, expounded in the author's own account, and reused and generalised by other authors (Gonçalves, Greenfeld and Madrid extended the method to a class of Collatz-like maps), with no objection to its correctness recorded in the literature. The result says nothing about every integer and does not imply the Collatz conjecture; a logarithmic-density-zero set of exceptions, which could still be infinite, is compatible with it. Whether the conclusion can be strengthened to natural density remains open.
Full reasoning: the evidence and decisions behind this verdict
The claim is Theorem 1.3 of Tao, "Almost all orbits of the Collatz map attain almost bounded values", arXiv:1909.03562 (September 2019), published as Forum of Mathematics, Pi 10 (2022), e12, doi:10.1017/fmp.2022.8 (www.cambridge.org/core/journals/forum-of-mathematics-pi/article/almost-all-orbits-of-the-collatz-map-attain-almost-bounded-values/1008CC2DF91AF87F66D190C5E01C907F). The three recorded sources (the refereed paper, the arXiv preprint, and Tao's blog announcement of 10 September 2019) are one author's voice, all affirming; the blog post states the theorem verbatim as its Theorem 2 and sketches the proof, while the paper carries it. The published abstract writes the bound as Col_min(N) ≤ f(N) and the theorem as Col_min(N) < f(N); the two are equivalent since f may be replaced by f minus 1. The stored text of the blog post drops its typeset formulas, so the theorem-statement passage recorded from it cannot be checked mechanically against that copy; it was checked against the paper instead.
Standing of the proof: the paper appeared in a selective refereed journal, Forum of Mathematics, Pi, and is open access; a web search found no published objection, erratum or retraction, and found a later paper (Gonçalves, Greenfeld and Madrid, "Generalized Collatz maps with almost bounded orbits", arXiv:2111.06170) that adapts the method to other maps, which is independent engagement with the argument by other mathematicians. The proof structure (reduction to stabilisation of a first-passage distribution for the Syracuse map, then to 3-adic fine-scale mixing, then to Fourier decay of a skew random walk controlled through a renewal process) is laid out in the paper's section headings and was read at the level of the introduction and reduction chain in this pass, not verified line by line. Under the domain's standards, a refereed, independently expounded proof that has stood without unresolved objection warrants "verified" as an accepted proof; there is no machine-checked proof, and none is expected soon given the length and analytic nature of the argument.
Weight of the supporting subclaim: the only attached subclaim, Korec's theorem, is prior evidence in the same direction rather than a premise; Tao's proof re-derives the local descent it needs. Neither theorem implies the other strictly, since Tao's is stated in logarithmic density and Korec's in natural density, so its assessment does not bear on this claim's status.
What would change the verdict: a published gap in the proof that survived scrutiny, or a counterexample to the stabilisation estimate; neither has appeared in the six years since announcement. Credence below 1 reflects only the general residual uncertainty of a long, unformalised analytic proof reviewed by a small number of referees.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Earlier results such as Korec's theorem that almost all N descend below N^θ for θ above log 3/log 4 give only local descent, which cannot be iterated directly because the Collatz dynamics do not carry the uniform distribution on one scale to the next. Tao's proof passes to the Syracuse map, builds an approximately invariant family of measures from the Syracuse random variables on Z/3^n Z, and reduces the theorem to a stabilisation property of the associated first-passage distribution, which in turn reduces to high-frequency decay of the characteristic function of a skew random walk on Z/3^n Z, proved through a two-dimensional renewal process escaping a union of well-separated triangles. Because this invariant-measure iteration turns local descent into descent below any function tending to infinity, the theorem follows for almost all N in logarithmic density.
The inference goes through: the paper's chain of reductions, from the theorem to stabilisation of the Syracuse first-passage distribution, to fine-scale 3-adic mixing, to Fourier decay of a skew random walk established via a renewal process, was refereed for Forum of Mathematics, Pi and has stood unchallenged since 2019, and later authors have adapted the same route to Collatz-like maps. The argument does not depend on Korec's descent theorem as a premise; that result marks the local-in-time state of the art the proof globalises, and the proof re-derives the descent it uses, so the argument's weight rests entirely on the paper's own analysis.
Provenance
Where this claim has been said, linked to its canonical form.
In this paper we obtain the following further improvement (at the cost of weakening natural density to logarithmic density):
Introduces Theorem 2: for any f with f(N)→+∞, Col_min(N) < f(N) for almost all N in the sense of logarithmic density.
The source's own evidence bears what it asserts. This sentence introduces the theorem rather than stating it; the stored text of the post drops its typeset formulas, so the theorem statement itself appears only in the paper.
Let {f: {\bf N}+1 \rightarrow {\bf R}} be any function with {\lim_{N \rightarrow \infty} f(N) = +\infty}. Then we have {\mathrm{Col}_{\min}(N) < f(N)} for almost all {N} (in the sense of logarithmic density).
Theorem 2, the paper's main result
The source's own evidence bears what it asserts. The blog post is the author's summary of his own paper: it states the theorem and outlines the proof but the evidence lives in the paper itself, which is the source to open for the argument. The quoted passage was not found in the stored copy of this source.
Let f : N+1 → R be any function with lim_{N→∞} f(N) = +∞. Then one has Col_min(N) < f(N) for almost all N ∈ N+1 (in the sense of logarithmic density).
Theorem 1.3, the main theorem of the refereed journal version (Forum of Mathematics, Pi, vol. 10, 2022, doi:10.1017/fmp.2022.8) of Tao's 2019 arXiv paper.
The source's own evidence bears what it asserts. This is the primary source: the refereed proof, published open access in Forum of Mathematics, Pi in 2022. The abstract states the bound with a weak inequality and the theorem with a strict one; since the function may be shifted by a constant, the two forms are equivalent. Worth reading closely: It contains the full proof; any doubt about the theorem's standing is settled by reading its reduction chain (first passage stabilisation, 3-adic mixing, Fourier decay). The quoted passage was not found in the stored copy of this source.
Theorem 1.3 (Almost all Collatz orbits attain almost bounded values). Let f : N+1 → R be any function with lim_{N→∞} f(N) = +∞. Then one has Col_min(N) < f(N) for almost all N ∈ N+1 (in the sense of logarithmic density).
Main theorem of the arXiv preprint later published in Forum of Mathematics, Pi.
How these sources relate
- https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ draws its statement from Almost all orbits of the Collatz map attain almost bounded values (arXiv:1909.03562), faithfully. The blog post is the author's own announcement of the arXiv paper and states the theorem in the same terms (any f tending to infinity, logarithmic density); no qualification is dropped or strengthened.
- Almost all orbits of the Collatz map attain almost bounded values is a later version of Almost all orbits of the Collatz map attain almost bounded values (arXiv:1909.03562). The Forum of Mathematics, Pi article (2022, doi:10.1017/fmp.2022.8) is the refereed published version of arXiv:1909.03562; same title, same author, same main theorem (Theorem 1.3).
- https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ and Almost all orbits of the Collatz map attain almost bounded values share an author. The blog post "Almost all Collatz orbits attain almost bounded values" (10 September 2019) is Terence Tao's own announcement of his paper; byline "by Terence Tao", and the post says "I've just uploaded to the arXiv my paper".
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.