I asked a related question before. Here I will ask the question as research and add details. In the first place, I am not sure that the problem I asked is the same / equivalent as Natural Density. If so, remember Terras' theorem.

Riho Terras (1979), On the existence of a density, Acta Arithmetica 35 (1979), 101-102. (MR 80h: 10066).

This document provides additional details about the evidence in Terras (1976) that

The set of integers that have an infinite detention time has a zero asymptotic density. The test in Terras (1976) had been criticized by Möller (1978). On an added note

As proof, the author states that the evidence of Terras (1976) is defective.

If the question I ask is really the same as Terras' theorem, then the answer is obvious: $$ lim_ {n to infty} dfrac { mathcal {C} (n)} { mathcal {N} (n)} = 1 $$

Now I want to ask my question.

Leave, $ mathcal {C} (n) $ Be a Collatz counting function. This function works like this:

Consider the exponential expansion interval: $ left (1, 2 ^ {2 ^ n}} – 1 right), n in mathbb {N}. $

The function $ mathcal {C} (n) $ count everything

Collatz numbersin the interval $ left (1,2 ^ 2 2 n) – 1 right) $. Here,Collatz numbersthey are odd positive integers when the function $$ mathrm {Coll (n)} = begin {cases} n / 2 & text {if} n equiv 0 pmod {2} \ 3n + 1 & text {if} n equiv 1 pmod {2}. end {cases} $$ applies to him, we definitely get the result $ 1 $. In other words, for big enough $ N $, we have $ mathrm {Coll ^ N (n)} = 1. $It is obvious that if there is a counterexample to the Collatz Conjecture, then we have countless counterexamples. Suppose we have

Finely many independent counterexamples.Or at least $ 1 $ verifiable counterexample. I define "independent verifiable counterexample" as follows:Leave, $ X $ Y $ Y $ Be verifiable counterexamples. Verifiable counterexamples are counterexamples that we can prove,

There is no such $ N $, what gives $ mathrm {Coll ^ N (X)} = 1 $ or $ mathrm {Coll ^ N (Y)} = 1. $ We work, yes and only if with odd positive integers. Independent verifiable counterexamples are counterexamples, for any $ N_1, N_2 in mathbb {N} $ we have $ mathrm {Coll ^ {N_1} (X)} neq mathrm {Coll ^ {N_2} (Y)} $. In other words, for $ X $ Y $ Y $we always have$ 2 $ Different trees / sequences.Leave, $$ displaystyle theta_1 = liminf_ {n to infty} dfrac { mathcal {C} (n)} { mathcal {N} (n)}, qquad theta_2 = limsup_ {n to infty} dfrac { mathcal {C} (n)} { mathcal {N} (n)} $$ where $ mathcal {N} (n) $ is the number of all odd positive integers in the interval $ left (1, 2 ^ 2 2 n) – 1 right) $ Y $ theta_1 leq theta_2 $. In this case, we have $ mathcal {N} (n) = 2 ^ 2 ^ n} -1} $

It is possible to show that, $ theta_1 $ strictly greater than $ 0. $

Suppose we have finitely many verifiable independent counterexamples (at least $ 1 $ verifiable counterexample), we can say $ theta_2 $ is strictly smaller than $ 1 $ ?

Of course, this is obvious if $ X $ verifiable counterexample, there are such $ n in mathbb {N} $ we have:

$$ liminf_ {n to N} dfrac { mathcal {C} (n)} { mathcal {N} (n)} = limsup_ {n to N} dfrac { mathcal {C} ( n)} { mathcal {N} (n)} <1 $$

Does this imply, if we have finely many independent verifiable counterexamples, or at least $ 1 $ verifiable counterexample then,

$$ theta_2 = limsup_ {n to infty} dfrac { mathcal {C} (n)} { mathcal {N} (n)} <1 $$

If, as I said before, the question is exactly the same as Terras 'theorem, then is Terras' theorem completely validated?

Note: I'm sorry for English grammars.

Thank you.