# Almost all orbits of the Collatz map attain almost bounded value

You need to know: Set ${\mathbb N}$ of positive integers, logarithm, limits.

Background: The Collatz map $\text{Col}: {\mathbb N}\to {\mathbb N}$ is defined by (i) $\text{Col}(n)=3n+1$ when $n$ is odd and (ii) $\text{Col}(n) = n/2$ when n is even. For any $n\in{\mathbb N}$, let $\text{Col}^k(n)$ denote the k-th iterate of $\text{Col},$ let $\text{Col}^{\mathbb N}(n)=\{n, \text{Col}(n), \text{Col}^2(n), \dots\}$ be the Collatz orbit, and let $\text{Col}_{\text{min}}(n) = \inf\limits_{k \in {\mathbb N}} \text{Col}^k(n)$  denote the minimal element of the Collatz orbit $\text{Col}^{\mathbb N}(n)$.

We say that subset $A\subseteq {\mathbb N}$ has logarithmic density 1, if $\lim\limits_{x\to \infty}\left(\frac{1}{\log x}\sum\limits_{k\in A, k\leq x}\frac{1}{k}\right)=1$. We say that a property P holds for almost all $n \in {\mathbb N}$ (in the sense of logarithmic density) if the subset of ${\mathbb N}$ for which P holds has logarithmic density 1.

The Theorem: On 8th September 2019, Terence Tao submitted to arxiv a paper in which he proved that for any function $f: {\mathbb N} \to {\mathbb N}$ such that $\lim\limits_{n\to \infty} f(n)=+\infty$, one has $\text{Col}_{\text{min}}(n) < f(n)$ for almost all $n \in {\mathbb N}$ (in the sense of logarithmic density).

Short context: The famous Collatz conjecture predicts that $\text{Col}_{\text{min}}(n)=1$ for all $n\in{\mathbb N}$, but it remains well beyond reach of the current methods. As a partial progress, Terras proved in 1976 that $\text{Col}_{\text{min}}(n) for almost all n. In 1979, Allouche improved this to $\text{Col}_{\text{min}}(n) for almost all n, for any fixed constant $\theta>0.869$. In 1994, Korec proved the same result for any $\theta>\frac{\log 3}{\log 4}=0.792...$. The Theorem implies that one can take any $\theta>0$, and, more generally, $\text{Col}_{\text{min}}(n) for almost all n and any function f going to infinity with n. For example, one can take $f(n)=\log\log\log\log n$.

Links: Free arxiv version of the original paper is here.

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