Almost all orbits of the Collatz map attain almost bounded values
(2019)cite arxiv:1909.03562Comment: 48 pages, 2 figures. Submitted, Forum of Math, Pi.

Define the Collatz map $Col : N+1 N+1$ on the positive integers $N+1 = \1,2,3,\dots\$ by setting $Col(N)$ equal to $3N+1$ when $N$ is odd and $N/2$ when $N$ is even, and let $Col_\min(N) := ınf_n N Col^n(N)$ denote the minimal element of the Collatz orbit $N, Col(N), Col^2(N), \dots$. The infamous Collatz conjecture asserts that $Col_\min(N)=1$ for all $N N+1$. Previously, it was shown by Korec that for any $> 34 0.7924$, one has $Col_\min(N) N^þeta$ for almost all $N ın N+1$ (in the sense of natural density). In this paper we show that for any function $f : N+1 R$ with $łim_N ınfty f(N)=+ınfty$, one has $Col_\min(N) f(N)$ for almost all $N ın N+1$ (in the sense of logarithmic density). Our proof proceeds by establishing an approximate transport 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 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.
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