%0 Journal Article %1 tao2019almost %A Tao, Terence %D 2019 %K mathematics theory %T Almost all orbits of the Collatz map attain almost bounded values %U http://arxiv.org/abs/1909.03562 %X 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.

@article{tao2019almost, 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.}, added-at = {2019-09-11T20:54:35.000+0200}, author = {Tao, Terence}, biburl = {https://www.bibsonomy.org/bibtex/20556cd6cd6ee92e0689448e8a6a59517/kirk86}, description = {[1909.03562] Almost all orbits of the Collatz map attain almost bounded values}, interhash = {281ba660189c287c7427b607da6a058d}, intrahash = {0556cd6cd6ee92e0689448e8a6a59517}, keywords = {mathematics theory}, note = {cite arxiv:1909.03562Comment: 48 pages, 2 figures. Submitted, Forum of Math, Pi}, timestamp = {2019-09-11T20:54:35.000+0200}, title = {Almost all orbits of the Collatz map attain almost bounded values}, url = {http://arxiv.org/abs/1909.03562}, year = 2019 }