The theory of the Collatz process and the method of dynamical balls

TL;DR

This paper develops a new theoretical framework using dynamical systems and dynamical balls to analyze the Collatz conjecture and its connections to prime distribution, offering novel formulations and tools.

Contribution

It introduces the theory of dynamical balls and their application to the Collatz process, providing new formulations and analytical tools for the conjecture.

Findings

New formulations of the Collatz conjecture in dynamical systems language

Development of tools to analyze convergence of sequences in the Collatz process

Establishment of connections between Collatz dynamics and prime distribution

Abstract

In this paper, we introduce and develop the theory of the Collatz process and the method of dynamical balls. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie Germain primes. We provide several formulations of the Collatz conjecture in this language. Furthermore, we introduce and develop the notion of dynamical systems induced by fixed $a\in \mathbb{N}$ and their associated induced dynamical balls. We develop tools to study problems that require determining the convergence of certain sequences generated by iterating on a fixed integer.