A non-uniform distribution property of most orbits, in case the $3x+1$ conjecture is true
Volume 178 / 2017
Abstract
Let $$ T(n)=\left\{\begin{array}{@{}ll@{}}3n+1&(n\hbox{ odd})\\ n/2&(n\hbox{ even})\end{array}\right.\quad\ (n\in\mathbb Z). $$ 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 write $c_i(n):=\#\{m\in\mathcal O_n:m\equiv i\bmod{18}\}$. Let $W$ be the set of integers whose orbit contains $1$ and is, in the following sense, approximately well distributed modulo $18$ between the six elements of the set $I:=\{1,5,7,11,13,17\}$ (the elements of $\{1,\ldots,18\}$ that are odd and not divisible by $3$). More precisely: $$ W:=\biggl\{n\in\mathbb Z:\exists k\ge0,\, T^k(n)=1\hbox{ and }\forall i\in I,\, \frac{c_i(n)}{\sum_{i\in I}c_i(n)}\le\frac16+0.0215\biggr\}. $$ We prove that $W\cap\mathbb N$ has density $0$ in $\mathbb N$. Consequently, if the $3x+1$ conjecture is true, most of the positive integers $n$ satisfy $$ \frac{\max_{i\in I}c_i(n)}{\sum_{i\in I}c_i(n)} \gt \frac16+0.0215. $$
