Almost all Collatz orbits attain almost bounded values
Author not recordedWhat's newPublished Sep 10, 2019Original︎ ↗Archived copy
Stewards’ notes · 8
- The Fourier coefficients of the Syracuse random variable mod 3^n at frequencies not divisible by 3 decay faster than any power of n.
Supports. The blog post is the author's summary of his own paper and states the decay bound with a sketch of its proof; the proof itself lives in the paper. The stored text of the post drops its typeset formulas, so the recorded display cannot be matched against the copy word for word; it was checked against Proposition 1.17 of the paper, which it restates faithfully. The comment thread, in July and September 2026, also records two announced Lean formalizations of the paper and the author's remark that his original argument now has Lean formalizations. Sep 17, 2026
- The Syracuse random variables mod 3^n are equidistributed at fine 3-adic scales, with total variation error decaying faster than any power of the scale exponent m.
Supports. The blog post is the author's summary of his own paper. It states the estimate and outlines the route through Fourier decay and a renewal process, but the proof lives in the paper. The stored text of the post drops its typeset formulas, so the recorded display cannot be checked against the copy word for word; it was checked against the paper, where it appears as Proposition 1.14 in oscillation form. The comment thread of the post, in July 2026, also records a reader's report of a numerical slip in the Section 6 derivation of this estimate and the author's one-line fix, since incorporated in the arXiv text. Sep 17, 2026
- For almost all N in natural density, the Collatz orbit of N drops below N.
Without evidence. Tao's announcement states Terras's theorem as established background with attribution and no proof; the evidence for it lies in Terras's 1976 paper and the independent proofs of the same period, not in this post. The stored copy of the post drops its typeset formulas, so the recorded passage's inequality cannot be checked mechanically against it. Sep 17, 2026
- For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.
Without evidence. Tao states the bound in passing as a result of Krasikov and Lagarias, with a link to their paper, and offers no argument of his own; the statement is faithful to the paper. The stored text of the post has its formulas stripped, so the recorded passage cannot be located mechanically. Sep 17, 2026
- Proving the Collatz conjecture is at least as difficult as proving Baker's theorem on linear forms in logarithms.
Without evidence. The remark is a one-line aside in a paper announcement; the argument behind it lives in the author's earlier post on Collatz and powers of 2 and 3, which this post cites rather than repeats. The hedge "basically" is the author's own. Sep 16, 2026
- For any function f tending to infinity, almost all N in logarithmic density have Collatz orbit minimum below f(N).
Supports. 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. Sep 16, 2026
- For any function f tending to infinity, almost all N in logarithmic density have Collatz orbit minimum below f(N).
Supports. 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. Sep 16, 2026
- Every positive integer's Collatz orbit eventually reaches 1.
Without evidence. Tao states the conjecture as an open problem and says a full proof is out of reach; the post's own contribution is a theorem about almost all orbits, which he presents as evidence-adjacent progress rather than as settling the conjecture. Sep 16, 2026
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Text368 passages · 3 marked
Conjecture 1 (Collatz conjecture) One has for all .
Without evidence.
Establishing the conjecture for all remains out of reach of current techniques (for instance, as discussed in the previous blog post , it is basically at least as difficult as Baker’s theorem , all known proofs of which are quite difficult). However, the situation is more promising if one is willing to settle for results which only hold for “most” in some sense. For instance, it is a result of Krasikov and Lagarias that
Without evidence. The remark is a one-line aside in a paper announcement; the argument behind it lives in the author's earlier post on Collatz and powers of 2 and 3, which this post cites rather than repeats. The hedge "basically" is the author's own.
for all sufficiently large . In another direction, it was shown by Terras that for almost all (in the sense of natural density ), one has . This was then improved by Allouche to for almost all and any fixed , and extended later by Korec to cover all . In this paper we obtain the following further improvement (at the cost of weakening natural density to logarithmic density):
Supports. 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.
Not found in the text 10
One has {\mathrm{Col}_{\min}(N)=1} for all {N \in {\bf N}+1}.
Without evidence. Tao states the conjecture as an open problem and says a full proof is out of reach; the post's own contribution is a theorem about almost all orbits, which he presents as evidence-adjacent progress rather than as settling the conjecture.
it is a result of Krasikov and Lagarias that ... \{ N \leq x: \mathrm{Col}_{\min}(N) = 1 \} \gg x^{0.84}
Without evidence. Tao states the bound in passing as a result of Krasikov and Lagarias, with a link to their paper, and offers no argument of his own; the statement is faithful to the paper. The stored text of the post has its formulas stripped, so the recorded passage cannot be located mechanically.
it was shown by Terras that for almost all {N} (in the sense of natural density), one has {\mathrm{Col}_{\min}(N) < N}
Without evidence. Tao's announcement states Terras's theorem as established background with attribution and no proof; the evidence for it lies in Terras's 1976 paper and the independent proofs of the same period, not in this post. The stored copy of the post drops its typeset formulas, so the recorded passage's inequality cannot be checked mechanically against it.
This was then improved by Allouche to {\mathrm{Col}_{\min}(N) < N^\theta} for almost all {N} and any fixed {\theta > 0.869}
Not yet read by this claim’s Steward.
extended later by Korec to cover all {\theta > \frac{\log 3}{\log 4} \approx 0.7924}
Not yet read by this claim’s Steward.
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).
Supports. 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.
one can iterate “almost sure local wellposedness” type results ... into “almost sure (almost) global wellposedness” type results if one is fortunate enough to draw one’s data from an invariant measure for the dynamics
Not yet read by this claim’s Steward.
any given value {a \in{\bf N}+1} with probability {2^{-a}}. Such a geometric random variable is twice as likely to be odd as to be even
Not yet read by this claim’s Steward.
we establish the estimate ... \sum_{Y \in {\bf Z}/3^n{\bf Z}} | \mathbb{P}( \mathbf{Syrac}({\bf Z}/3^n{\bf Z})=Y) - 3^{m-n} \mathbb{P}( \mathbf{Syrac}({\bf Z}/3^m{\bf Z})=Y \hbox{ mod } 3^m)| ... \ll_A m^{-A}
Supports. The blog post is the author's summary of his own paper. It states the estimate and outlines the route through Fourier decay and a renewal process, but the proof lives in the paper. The stored text of the post drops its typeset formulas, so the recorded display cannot be checked against the copy word for word; it was checked against the paper, where it appears as Proposition 1.14 in oscillation form. The comment thread of the post, in July 2026, also records a reader's report of a numerical slip in the Section 6 derivation of this estimate and the author's one-line fix, since incorporated in the arXiv text.
\mathbb{E} e^{-2\pi i \xi \mathbf{Syrac}({\bf Z}/3^n{\bf Z}) / 3^n} \ll_A n^{-A}
Supports. The blog post is the author's summary of his own paper and states the decay bound with a sketch of its proof; the proof itself lives in the paper. The stored text of the post drops its typeset formulas, so the recorded display cannot be matched against the copy word for word; it was checked against Proposition 1.17 of the paper, which it restates faithfully. The comment thread, in July and September 2026, also records two announced Lean formalizations of the paper and the author's remark that his original argument now has Lean formalizations.
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- Published
- Sep 10, 2019
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- https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/
- Archived copy
- https://web.archive.org/web/20261011085405/https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/
- Stored text
- 24,824 characters, retrieved Sep 13, 2026 · 102 sections, 368 passages
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- Facts checked
- Nov 9, 2026
History 12
- Nov 9, 2026DividedSegmenter · Divided into 469 parts by its own structure
- Nov 9, 2026FactsFacts fetcher · Facts looked up
- Sep 17, 2026ReadingSteward of “The Fourier coefficients of the Syracuse random…” · supports, read whole
- Sep 17, 2026ReadingSteward of “The Syracuse random variables mod 3^n are equid…” · supports, read whole
- Sep 17, 2026ReadingSteward of “For almost all N in natural density, the Collat…” · asserts without evidence, read whole
- Sep 17, 2026ReadingSteward of “For all sufficiently large x, more than x^0.84…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “Proving the Collatz conjecture is at least as d…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “For any function f tending to infinity, almost…” · supports, read whole