Almost all orbits of the Collatz map attain almost bounded values

Abstract

Define the \emph{Collatz map} $\mathrm{Col} : \mathbb{N}+1 \to \mathbb{N}+1$ on the positive integers $\mathbb{N}+1 = \{1,2,3,\dots\}$ by setting $\mathrm{Col}(N)$ equal to $3N+1$ when $N$ is odd and $N/2$ when $N$ is even, and let $\mathrm{Col}_{\min}(N) := \inf_{n \in \mathbb{N}} \mathrm{Col}^n(N)$ denote the minimal element of the Collatz orbit $N, \mathrm{Col}(N), \mathrm{Col}^2(N), \dots$. The infamous \emph{Collatz conjecture} asserts that $\mathrm{Col}_{\min}(N)=1$ 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 $\mathrm{Col}_{\min}(N) \leq N^\theta$ for almost all $N \in \mathbb{N}+1$ (in the sense of natural density). In this paper we show that for \emph{any} function $f : \mathbb{N}+1 \to \mathbb{R}$ with $\lim_{N \to \infty} f(N)=+\infty$, one has $\mathrm{Col}_{\min}(N) \leq f(N)$ for almost all $N \in \mathbb{N}+1$ (in the sense of logarithmic density). Our proof proceeds by establishing 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.