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Statement

Setting (pp. 9--10). For a positive integer $n$ the total stopping time is $\sigma_\infty(n)=\inf\{k\ge0:T^{(k)}(n)=1\}$, with $\sigma_\infty(n)=+\infty$ when no such $k$ exists (Definition 2.3, p. 9), where $T$ is the $3x+1$ function. Definition 2.4 (p. 10) sets $\gamma_\infty(n)=\sigma_\infty(n)/\log n$ for $n\ge1$, and Definition 2.5 (p. 10) defines the $3x+1$ scaled stopping constant

$$\gamma=\gamma_3:=\limsup_{n\to\infty}\gamma_\infty(n)=\limsup_{n\to\infty}\frac{\sigma_\infty(n)}{\log n}.$$

The model constant (Theorem 4.1, p. 24, credited to Lagarias and Weiss). In the $3x+1$ biased random walk the steps are $\log\frac32$ and $\log\frac12$, each with probability $\frac12$, and $S_\infty(n)$ is the first $k>0$ with $Z_k\le0$ for the walk started at $Z_0=\log n$ (p. 20). The repeated random walk (RRW) model runs one independent such walk for each $n\ge1$ (§4.1, p. 23). Theorem 4.1 states that with probability one $\limsup_{n\to\infty}S_\infty(n)/\log n$ is finite and equals a constant $\gamma_{RRW}\approx41.677647$, the unique real $\gamma>\left(\frac12\log\frac43\right)^{-1}\approx6.952$ with $\gamma\,g(1/\gamma)=1$, where $g(a)=\sup_{\theta\in\mathbb R}(\theta a-\log M_{RRW}(\theta))$ and $M_{RRW}(\theta)=\frac12\left(2^\theta+(2/3)^\theta\right)$.

Conjecture 4.1 (p. 25, $3x+1$ Scaled Stopping Constant Conjecture), quoted: "The $3x+1$ scaled stopping constant $\gamma$ is finite and is given by $\gamma=\gamma_{RRW}\approx 41.677647$."

The survey adds (p. 25) that the model also predicts the shape of near-extremal trajectories: in the scaling $(k/\log n,\ \log T^{(k)}(n)/\log n)$ they should follow the segment from $(0,1)$ to $(\gamma_{RRW},0)$. In §4.4 (pp. 26--27) it notes that the RRW model ignores the coalescence of real trajectories, and rests its confidence in the conjecture on the branching random walk model giving the same constant (Theorem 6.4) and on the record data of its Table 1. The introduction (p. 4) restates the prediction: only finitely many trajectories starting at $x$ need more than $(\gamma_3+\epsilon)\log x$ steps to reach $1$, and infinitely many need more than $(\gamma_3-\epsilon)\log x$, with $\gamma_3\approx41.67765$.

Source. A. V. Kontorovich and J. C. Lagarias, Stochastic models for the $3x+1$ and $5x+1$ problems, arXiv:0910.1944v1 (2009), 66 pp.; published in The Ultimate Challenge: The $3x+1$ Problem (AMS, 2010). Pages and labels are those of the arXiv v1 print; the edition read is identified on the source card.

Proof pointer

None: the statement is a conjecture. Theorem 4.1 is Lagarias and Weiss, The $3x+1$ problem: two stochastic models, Ann. Appl. Probab. 2 (1992), 229--261, Theorem 2.1, as the survey cites it (p. 23); the survey gives no proof.

Read depth

Claims checked: Definitions 2.3 to 2.5, Theorem 4.1 and Conjecture 4.1 were read clause by clause on the page images of the print. The proof of Theorem 4.1 is not in the survey and was not read. Nothing here is independently reviewed.

Dependencies

Theorem 4.1 (Lagarias and Weiss), for the value of $\gamma_{RRW}$.

Bears on

• Problem 1135: the problem's $f$ is the survey's $T$ on the positive integers. If some positive $m$ never reaches $1$, then neither does any $2^jm$, since $T(2n)=n$; so $\gamma_\infty(n)=+\infty$ for infinitely many $n$ and $\gamma=+\infty$. Hence the finiteness part of Conjecture 4.1 implies an affirmative answer to the problem, and the full conjecture adds $\sigma_\infty(n)\le(\gamma_{RRW}+o(1))\log n$. This deduction is the corpus's; the survey remarks only (p. 10) that $\gamma_\infty(n)$ is finite for all positive $n$ only if the $3x+1$ conjecture is true. The conjecture is unproved, and the survey gives it heuristic support only.