Under Collatz conjecture the Collatz mapping has no an asymptotic mixing property $\pmod{3}$

TL;DR

This paper investigates the distribution of even numbers in Collatz sequences using Markov chains and measure theory, showing the lack of asymptotic mixing modulo 3 and linking the conjecture to measure-theoretic properties.

Contribution

It introduces a measure-theoretic approach to analyze Collatz sequences and demonstrates the absence of asymptotic mixing modulo 3, providing new insights into the conjecture's structure.

Findings

Relative frequency of even numbers approaches 2/3 with oscillations.

Numerical characteristics for numbers of the form 3m+1 are explicitly derived.

The paper links Collatz conjecture to measure-theoretic properties and supernatural numbers.

Abstract

By using properties of Markov homogeneous chains and Banach measure in $\mathrm{N}$, it is proved that a relative frequency of even numbers in the sequence of $n$-th coordinates of all Collatz sequences is equal to the number $\frac{2}{3}+\frac{(-1)^{n+1}}{3\times 2^{n+1}}.$ It is shown also that an analogous numerical characteristic for numbers of the form $3m+1$ is equal to the number $\frac{3}{5}+ \frac{(-1)^{n+1}}{15 \times 2^{2(n-1)}}. $ By using these formulas it is proved that under Collatz conjecture the Collatz mapping has no an asymptotic mixing property $\pmod{3}$. It is constructed also an example of a real-valued function on the cartesian product $N^2$ of the set of all natural numbers $N$ such that an equality its repeated integrals (with respect to Banach measure in $N$) implies that Collatz conjecture fails. In addition, it is demonstrated that Collatz conjecture fails for supernatural numbers.