Tao 2019 almost all orbits

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theorem_1_3: Tao's theorem that for every function f tending to infinity the minimal value of the Collatz orbit of N is below f(N) for almost all N in the sense of logarithmic density; the strongest partial result on the Collatz conjecture recorded on Problem 1135, stated for the standard map and transferred to the shortcut map by the paper's own remark.

theorem_1_6: The Syracuse form of Tao's main theorem: for every function f on the odd positive integers tending to infinity, the minimal value of the Syracuse orbit of N is below f(N) for almost all odd N in logarithmic density; the paper shows it implies Theorem 1.3 and states the two are equivalent.

theorem_3_1: The quantitative form of Tao's main theorem: for N_0 >= 2 and x >= 2 the logarithmically weighted proportion of odd N up to x whose Syracuse orbit stays above N_0 is O(1/log^c N_0), uniformly in x, and likewise for the Collatz orbits of all positive N up to x.

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Terence Tao, Almost All Orbits of the Collatz Map Attain Almost Bounded Values, Forum Math. Pi 10 (2022), Paper No. e12, 56 pp.; DOI 10.1017/fmp.2022.8 (published online 20 May 2022; the Crossref record). The copy read for this card is arXiv:1909.03562v7 [math.PR] (16 July 2026), 58 pages with a complete text layer; the arXiv record carries the journal reference above. The journal text was not compared; the locators below are the preprint's. The statement pages were read on the rendered page images of pp. 1--4 and 13--18. Source: PDF. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1909.03562), every other right reserved.

Read status: claims checked for the abstract, Conjecture 1.1, the partial results paragraph, Theorem 1.3, Remark 1.4 and the Section 1.2 remark on the accelerated map, read clause by clause on the page images of pp. 1--3; Conjecture 1.5, Theorem 1.6 and identity (1.2) on p. 4, and Theorem 3.1 on pp. 16--17, read clause by clause on the page images; the proofs (Sections 2--7) were not read.

Relevance: Proves almost all Collatz orbits attain almost bounded values in logarithmic density, the strongest known partial result on problem 1135.

Contents

• Section 1.1 (p. 1): the Collatz map $\mathrm{Col}(N)=3N+1$ ($N$ odd), $N/2$ ($N$ even) on $\mathbb N+1=\{1,2,\ldots\}$; $\mathrm{Col}_{\min}(N)=\min\mathrm{Col}^{\mathbb N}(N)$, the minimal element of the orbit; Conjecture 1.1 (Collatz conjecture): $\mathrm{Col}_{\min}(N)=1$ for all $N\in\mathbb N+1$. "While the full resolution of Conjecture 1.1 remains well beyond reach of current methods, some partial results are known": numerical verification for all $N\le5.78\times10^{18}$ [17], $N\le10^{20}$ [18] and "most recently for all $N\le2^{68}\approx2.95\times10^{20}$ [3]" (Barina), and Krasikov and Lagarias's $\#\{N\le x:\mathrm{Col}_{\min}(N)=1\}\gg x^{0.84}$ [13] (pp. 1--2). Definition 1.2 (p. 2) defines almost all $N$ through logarithmic density, and p. 2 recalls the earlier almost-all bounds, which also hold in natural density: $\mathrm{Col}_{\min}(N)<N$ (Terras, independently Everett), and $\mathrm{Col}_{\min}(N)<N^\theta$ for every fixed $\theta$ above a constant $\approx0.869$ (Allouche), extended to $\theta>\frac{\log3}{\log4}\approx0.7924$ (Korec). The closed form printed for Allouche's constant, $\frac32-\frac{\log3}{\log2}$, is negative; the printed value $0.869$ is $\frac32-\frac{\log2}{\log3}$ to three places.
• Theorem 1.3 (p. 3): for any $f:\mathbb N+1\to\mathbb R$ with $\lim_{N\to\infty}f(N)=+\infty$, $\mathrm{Col}_{\min}(N)<f(N)$ for almost all $N\in\mathbb N+1$ in the sense of logarithmic density; "Thus for instance one has $\mathrm{Col}_{\min}(N)<\log\log\log\log N$ for almost all $N$." Remark 1.4: a bounded constant $C_0$ in place of $f(N)$ "is likely to be almost as hard to settle as the full Collatz conjecture"; the theorem is equivalent to: for any $\delta>0$ there is $C_\delta$ with $\mathrm{Col}_{\min}(N)\le C_\delta$ on a set of lower logarithmic density at least $1-\delta$, with $C_\delta\ll\exp(\delta^{-O(1)})$ (Theorem 3.1).
• Section 1.2 (p. 3): the accelerated map $\mathrm{Col}_2(N)=(3N+1)/2$ ($N$ odd), $N/2$ ($N$ even), under which every step divides by $2$ just once; "It is easy to see that $\mathrm{Col}_{\min}(N)=(\mathrm{Col}_2)_{\min}(N)$ for all $N\in\mathbb N+1$, so all the results in this paper concerning $\mathrm{Col}$ may be equivalently reformulated using $\mathrm{Col}_2$." The proof itself uses the Syracuse map (one multiplication by $3$ per step) and 3-adic analysis.
• Section 1.2 (p. 4): the Syracuse map $\mathrm{Syr}$ on the odd positive integers $2\mathbb N+1$, sending $N$ to the largest odd divisor of $3N+1$; identity (1.2), $\mathrm{Col}_{\min}(N)=\mathrm{Syr}_{\min}(N/2^{\nu_2(N)})$; Conjecture 1.5 (the Syracuse formulation, $\mathrm{Syr}_{\min}(N)=1$ for all odd $N$); and Theorem 1.6, the Syracuse form of Theorem 1.3, which the paper states is equivalent to it and from which it deduces Theorem 1.3.
• Theorem 3.1 (pp. 16--17), the alternate form of the main theorem: for $N_0\ge2$ and $x\ge2$ the logarithmic proportion of odd $N\le x$ with $\mathrm{Syr}_{\min}(N)>N_0$ is $\ll\log^{-c}N_0$, and likewise $\mathrm{Col}_{\min}(N)\le N_0$ for all but $O(\log^{-c}N_0)$ of $\mathbb N+1\cap[1,x]$ in logarithmic measure; it implies Theorem 1.6 (p. 18).
• Sections 2--7 (pp. 13--56), outlined in Sections 1.2--1.4 (pp. 3--13): the proof, through a stabilization property of a first-passage random variable for the Syracuse iteration and estimates for the characteristic function of a skew random walk on $\mathbb Z/3^n\mathbb Z$ (abstract; not read).

Compiled scope

The introduction's statements were read on the page images; Theorems 1.3, 1.6 and 3.1 are compiled as statements with the paper's proof pointers. No step of the proof was read or checked and nothing here is independently reviewed.

Bears on. #1135: the strongest partial result recorded on the page, for almost all starting values in logarithmic density; stated for the standard Collatz map, and carried to the page's shortcut map by the paper's own remark that the orbit minima of the two maps coincide (Section 1.2). It does not decide any single starting value and leaves the conjecture open, as the paper says. Theorem 1.6 is the same result for the Syracuse map, from which the paper deduces Theorem 1.3, and Theorem 3.1 is its quantitative form: for $N_0\ge2$ and $x\ge2$, all but a logarithmic proportion $O(\log^{-c}N_0)$ of $N\le x$ have orbit minimum at most $N_0$, with $c$ an absolute constant not made explicit.

Results.

• Theorem 1.3 (p. 3): $\mathrm{Col}_{\min}(N)<f(N)$ for almost all $N$ in logarithmic density, for every $f$ tending to infinity.
• Theorem 1.6 (p. 4): $\mathrm{Syr}_{\min}(N)<f(N)$ for almost all odd $N$, for every $f$ on the odd positive integers tending to infinity.
• Theorem 3.1 (pp. 16--17): the logarithmic proportion of odd $N\le x$ with $\mathrm{Syr}_{\min}(N)>N_0$ is $\ll\log^{-c}N_0$, for $N_0,x\ge2$.

No file of this source is held: no license on record permits redistribution of the edition read, and the card cites that edition. The Crossref record lists CC BY 4.0 for the journal version, an edition not read here.