Density of Collatz Trajectories

TL;DR

This paper introduces a new method to analyze the densities of Collatz trajectories, providing an alternative proof for a key density result and discussing potential improvements for related conjectures.

Contribution

The paper presents a novel approach to studying Collatz trajectory densities and offers an alternative proof for a significant density property.

Findings

Established that the density of numbers with trajectories eventually below a linear bound is 1.

Proposed a new method for analyzing densities of Collatz trajectories.

Discussed potential improvements for density results involving polynomial bounds.

Abstract

In this paper I present a new method of studying the densities of the Collatz trajectories generated by a set $S \subset \mathbb{N}$. This method is used to furnish an alternative proof that $d(\{y \in \mathbb{N} : \exists k \text{ where } T^k(y) < cy\}) = 1$ for all $c > 0$. Finally, I briefly discuss how the ideas presented in this paper could be used to improve the result that $d(\{y \in \mathbb{N} : \exists k \text{ where } T^k(y) < y^c\}) = 1$ for $c > \log_4 3$.