Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Under Collatz conjecture the Collatz mapping has no an asymptotic mixing
  property $\pmod{3}$   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.

14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
3
The categories of this paper are: physics