Generalized Collatz Maps with Almost Bounded Orbits

TL;DR

This paper generalizes Tao's recent result on the Collatz conjecture by showing that for a broad class of Collatz-like maps, almost all orbits become bounded or attain values below any given growth function, under certain conditions.

Contribution

It extends Tao's result to a wider class of Collatz-like maps, demonstrating that almost all orbits are almost bounded under suitable conditions on parameters.

Findings

Almost all orbits attain values below any prescribed growth function.

Generalization of Tao's result to maps with different parameters p and q.

Under certain conditions, orbits exhibit almost bounded behavior.

Abstract

If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.