Almost all orbits of the Collatz map attain almost bounded values (1909.03562v5)

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.

• The paper’s main contribution is proving that almost all integers’ Collatz orbits fall below any unbounded, slowly growing function in a logarithmic density sense.
• It employs an accelerated Syracuse map and geometric random variables to analyze the 2-adic valuation sequences during iterations.
• The results uncover hidden regularity in chaotic dynamics, opening avenues for further exploration of the Collatz conjecture.

An Overview of Terence Tao's Work on the Collatz Conjecture

Terence Tao's paper examines the infamous Collatz conjecture through a probabilistic and dynamical systems approach. The Collatz map is defined on the set of positive integers such that if an integer $N$ is odd, it maps to $3N + 1$, and if $N$ is even, it maps to $N/2$. The Collatz conjecture hypothesizes that, starting from any positive integer and iterating the map, one will ultimately reach the integer 1. Despite its simplicity, this problem has remained unsolved for decades and presents significant challenges in number theory.

Main Results

Tao's central result is the demonstration that, for any function $f: \mathbb{N} + 1 \to \mathbb{R}$ with the property that $\lim_{N \to \infty} f(N) = +\infty$, almost all integers $N$ will have Collatz orbits with elements dropping below $f(N)$. This result holds in the sense of logarithmic density, a measure that better captures multiplicative invariance properties necessary for Tao's analysis.

This work builds on prior results of establishing upper bounds on the minimal values attained by the Collatz orbit for almost all integers. Previous results by Korec showed that such bounds exist for $_{\min}(N) \leq N^\theta$ with $\theta > \frac{\log 3}{\log 4}$. Tao's advancement allows the bound to be replaced with any slowly growing function $f(N)$ tending to infinity.

Methodology

Tao leverages a combination of probabilistic and analytic tools, focusing on an accelerated form of the Collatz map known as the Syracuse map, which maps odd integers $N$ to the largest odd divisor of $3N + 1$. This formulation restricts the behavior of iterates to odd numbers, offering a more tractable structure for analysis. The author deploys both geometric and concentration inequalities to explore the statistical distribution of Syracuse map iterations.

A significant component of Tao's approach is the analysis of the $n$-Syracuse valuations—a tuple describing the 2-adic valuation sequence during the iterations. Tao models these valuations through geometric random variables, drawing on heuristics and total variation bounding to justify their statistical treatment.

An instrumental part of the paper is the stabilization property involving first passage times of the Syracuse map. Tao constructs a family of probability measures invariant under the passage map, ensuring that the iterated dynamics of starting points attain values close to bounded states with high probability.

Probabilistic and Dynamical Insights

The analysis extensively utilizes probabilistic renewal processes and renewal rewards to describe the dynamic steps of the map iterations, embedding the problem within a dynamical systems framework. Importantly, the paper uncovers a "fine-scale mixing" property—an approximate uniform distribution achieved at high frequencies—complemented by strong decay bounds on characteristic functions of associated Syracuse random variables. The exploration reveals that most trajectories, under minor restrictions on the starting distributions, exhibit substantial regularity in their valuation decomposition.

Implications and Future Directions

Although Tao's results do not resolve the entire Collatz conjecture, they significantly expand our understanding of the behavior of orbits on a logarithmic scale for 'almost all' integers. These insights hint at the inherent 'chaotic' yet regular underpinnings within what seems an impenetrably irregular sequence of operations.

The strategic combination of number theory, dynamical systems, and probability theory in his approach may lay the groundwork for exploring further inherent arithmetic structures in unsolved problems alike. Tao mentions future potential to refine these techniques towards addressing the Collatz conjecture on a broader range of densities, including natural density, which would mark a significant stride in this domain. Future developments may also capitalize on enhanced computational strategies or deeper exploration into the probabilistic laws governing these unexpected regularities, perhaps leading to insights in neighboring conjectures or unexplored territory in integer dynamics.