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I have recently managed to show a sequence of sufficiency sets for the Collatz conjecture whose natural density approaches 0 (the set theoretic limit approaches the set $\{1\}$). It is an extension of the result of Andaloro where he has shown that the Collatz conjecture is true iff it holds for all number congruent to 1 modulo 16 (modulo 2 and 4 is obvious and modulo 8 he has shown in the same paper). After a little work I was able to show the same characterization for any chosen power of 2 i.e.: Given any natural number n, Collatz conjecture is true iff it holds for all numbers congruent to 1 modulo $2^n$. Well, one is already intuitively thinking of this extension after reading Andaloro.

Personally I do not think that this is a major achievement yet but I was wondering if anyone has come up with this already? I would gladly share a preprint in arXiv but apparently I need someone in this field who can endorse this (I don't recall this procedure in the past when I was submitting to arXiv). I saw from an article by Marc Chamberland that the nearest result to this is the paper by Korec and Znám where they reduced the Collatz problem to sufficiency sets of the form $$\{x : x\equiv a\mod p^n\}$$ where $n$ is any chosen natural number, $p$ is an odd prime and $a$ is a primitive root modulo $p^2$ such that $p\not\mid a$. One can choose $n$ large enough to have a very small natural density for this set. Well for this natural generalisation of Andaloro's result, I think that the conditions are much easier and one can similarly choose a set from the sequence of sets that has an arbitrarily small natural density.

I don't have much exposure to this area of research yet so I want to know if this is anything of great relevance for researchers in this area?

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  • $\begingroup$ I added the [reference-request] tag. $\endgroup$
    – David Roberts ♦
    Commented Jul 23, 2015 at 6:27
  • $\begingroup$ The real problem is to show that the Collatz conjecture holds for all numbers in some residue class -- this is somehow "orthogonal" to your result. $\endgroup$
    – Stefan Kohl ♦
    Commented Jul 23, 2015 at 20:46
  • $\begingroup$ By the way -- Andaloro's result is pretty straightforward, see here. $\endgroup$
    – Stefan Kohl ♦
    Commented Jul 23, 2015 at 20:47
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    $\begingroup$ Do you have much experience of p-adics and finite-tailed Laurent series? $\endgroup$
    – Robert Frost
    Commented Mar 8, 2017 at 14:18

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Based on my (amateur) research, what would be of particular interest with respect to your proof is the question of whether you can extend it into the $2$-adic metric space and show that the same holds for the limit in the $2$-adic metric space of $1 \mod 2^n$ as $n\to\infty$. Which would of course, prove the conjecture.

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