A Collatz-type conjecture on the set of rational numbers

TL;DR

This paper explores a Collatz-like conjecture on positive rational numbers, proposing that all such numbers eventually reach zero under a specific transformation, and proves that the set of elements with rational fixed points is negligible.

Contribution

It introduces a new Collatz-type conjecture for rational numbers and proves that the set of elements with rational fixed points has asymptotic density zero.

Findings

The conjecture that all positive rationals end in zero remains open.

The set of elements with rational fixed points has asymptotic density zero.

No positive rational fixed points exist for the generated maps.

Abstract

Define $\theta(x)=(x-1)/3$ if $x\geq 1$, and $\theta(x)=2x/(1-x)$ if $x<1$. We conjecture that the orbit of every positive rational number ends in 0. In particular, there does not exist any positive rational fixed point for a map in the semigroup $\Omega$ generated by the maps $3x+1$ and $x/(x+2)$. In this paper, we prove that the asymptotic density of the set of elements in $\Omega$ that have rational fixed points is zero.