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Statement

Setting (p. 17). $T$ is the $3x+1$ function, $T(n)=(3n+1)/2$ for odd $n$ and $T(n)=n/2$ for even $n$, on all integers ((1.2), p. 2). For an integer $a$, Definition 2.10 sets $\pi_a(x)$ to be the number of integers $n$ with $|n|\le x$ such that $T^{(k)}(n)=a$ for some $k\ge0$. Definition 2.11 sets

$$\eta_3^+(a)=\limsup_{x\to\infty}\frac{\log\pi_a(x)}{\log x},\qquad \eta_3^-(a)=\liminf_{x\to\infty}\frac{\log\pi_a(x)}{\log x},$$

and, when the two agree, calls the common value the $3x+1$ growth exponent $\eta_3(a)$. For $a\equiv0\pmod 3$ the inverse orbit of $a$ is $\{2^ka:k\ge0\}$, so $\eta_3(a)=0$.

Conjecture 2.1 (p. 17, $3x+1$ Growth Exponent Conjecture), quoted: "For all integers $a\not\equiv 0 \pmod 3$, the $3x+1$ growth exponent $\eta_3(a)$ exists, with $\eta_3(a)=1$."

The survey attributes the conjecture to Applegate and Lagarias, and adds (p. 18) that the $3x+1$ Conjecture would imply $\eta_3(1)=1$ but does not seem to determine $\eta_3(a)$ for every such $a$; that Applegate and Lagarias also conjectured the stronger linear bound $\pi_a(x)>c_ax$ for all $x\ge1$, with a constant $c_a>0$; that Theorem 2.5 (Krasikov and Lagarias: $\pi_a(x)\ge x^{0.84}$ for $x\ge x_0(a)$, p. 17) gives $\eta_3^-(a)\ge0.84$; and that the branching random walk model of §6.5 predicts $\eta_3(a)=1$ (see Theorem 6.5).

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. The lower bound behind $\eta_3^-(a)\ge0.84$ is Krasikov and Lagarias, Acta Arith. 109 (2003); the survey says the exponent was computed with $k=9$ (p. 17), while that paper's own result page, Theorem 6.1, records a computation with $k=11$.

Read depth

Claims checked: Definitions 2.10 and 2.11, Conjecture 2.1 and the remarks on p. 18 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Theorem 2.5 of the survey (Krasikov and Lagarias 2003), as reported, for the lower bound $0.84$.

Bears on

• Problem 1135: the problem's $f$ is the survey's $T$ on the positive integers. An affirmative answer to the problem puts every positive integer up to $x$ into the count $\pi_1(x)$, and $\pi_1(x)\le2x+1$, so it gives $\eta_3(1)=1$, as the survey notes (p. 18). The converse does not follow: $\eta_3(1)=1$, or Conjecture 2.1 for every $a$, leaves room for positive integers that never reach $1$, so neither would answer the problem. The survey proves neither.