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.
Categories: math.PR math.DS math.NT