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

Abstract

Let $T(n)=\{\begin{array}{llc}3n+1& (nodd)\frac{n}{2}& (neven)\end{array}$ ($n\in \mathbb{Z}$). We call "the orbit of the integer $n$", the set $${\mathcal{O}}_n:=\{m\in \mathbb{Z}:\exists k\ge 0,m=T^k(n)\}$$ and we put $c_i(n):=\#\{m\in {\mathcal{O}}_n:m\equiv imod.18\}$. 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: $$W:=\{n\in \mathbb{N}:\exists k\ge 0,T^k(n)=1\; and\; \forall i\in I,\frac{c_i(n)}{{\sum }_{i\in I}{c}_i(n)}\le \frac{1}{6}+0.0215\}.$$ 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 $$\frac{{\max }_{i\in I}{c}_i(n)}{{\sum }_{i\in I}{c}_i(n)}>\frac{1}{6}+\mathrm{0.0215.}$$

9 pages, to publish in Acta Arithmetica