Almost all orbits of the Collatz map attain almost bounded values
Abstract Define the Collatz map 
${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$
 on the positive integers 
$\mathbb {N}+1 = \{1,2,3,\dots \}$
 by setting 
${\operatorname {Col}}(N)$
 equal to 
$3N+1$
 when N is odd and 
$N/2$
 when N is even, and let 
${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$
 denote the minimal element of the Collatz orbit 
$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $
 . The infamous Collatz conjecture asserts that 
${\operatorname {Col}}_{\min }(N)=1$
 for all 
$N \in \mathbb {N}+1$
 . Previously, it was shown by Korec that for any 
$\theta> \frac {\log 3}{\log 4} \approx 0.7924$
 , one has 
${\operatorname {Col}}_{\min }(N) \leq N^\theta $
 for almost all 
$N \in \mathbb {N}+1$
 (in the sense of natural density). In this paper, we show that for any function 
$f \colon \mathbb {N}+1 \to \mathbb {R}$
 with 
$\lim _{N \to \infty } f(N)=+\infty $
 , one has 
${\operatorname {Col}}_{\min }(N) \leq f(N)$
 for almost all 
$N \in \mathbb {N}+1$
 (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a 
$3$
 -adic cyclic group 
$\mathbb {Z}/3^n\mathbb {Z}$
 at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.