# Publishing house / Journals and Serials / Acta Arithmetica / All issues

## Acta Arithmetica

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

### Volume 178 / 2017

Acta Arithmetica 178 (2017), 125-134 MSC: 11B37, 11A99, 11B83. DOI: 10.4064/aa8385-9-2016 Published online: 23 March 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 $$\frac{\max_{i\in I}c_i(n)}{\sum_{i\in I}c_i(n)} \gt \frac16+0.0215.$$

#### Authors

• Alain Thomas448 allée des Cantons
83640 Plan-d’Aups-Sainte-Baume, France
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

## Rewrite code from the image 