A non-uniform distribution property of most orbits, in case the $3x+1$ conjecture is true

TL;DR

This paper investigates the distribution of orbits in the Collatz conjecture, showing that most integers do not have well-distributed orbits modulo 18 if the conjecture holds, indicating a non-uniform distribution property.

Contribution

It introduces a new measure of orbit distribution modulo 18 and proves that the set of integers with well-distributed orbits has density zero, assuming the Collatz conjecture is true.

Findings

Most integers do not have well-distributed orbits modulo 18.

The set of integers with well-distributed orbits has density zero.

If the Collatz conjecture is true, the orbits are generally non-uniformly distributed.

Abstract

Let $T(n)=\left\{\begin{array}{ll}3n+1&(n\hbox{ odd})\frac n2&(n\hbox{ even})\end{array}\right.$ ($n\in\mathbb Z$). We call "the orbit of the integer $n$", the set $$ \mathcal O_n:=\{m\in\mathbb Z\;:\;\exists k\ge0,\ m=T^k(n)\} $$ and we put $c_i(n):=\#\{m\in\mathcal O_n\;:\;m\equiv i\hbox{ mod.}18\}$. Let $W$ be the set of the integers whose orbit contains $1$ and is, in the following sense, about well distributed modulo $18$ between the six elements of the set $I:=\{1,5,7,11,13,17\}$ (the elements of \{1,\dots,18\} that are odd and not divisible by $3$). More precisely: $$ W:=\Big\{n\in\mathbb N\;:\;\exists k\ge0,\ T^k(n)=1\hbox{ and }\forall i\in I,\ \frac{c_i(n)}{\sum_{i\in I}c_i(n)}\le\frac16+0.0215\Big\}. $$ We prove that $W$ has density $0$ in $\mathbb N$. Consequently, if the $3x+1$ conjecture is true, most of the positive integers $n$ satisfy $$ \frac{\max_{i\in I}c_i(n)}{\sum_{i\in I}c_i(n)}>\frac16+0.0215. $$