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I asked this question on MSE. It received up-votes and one answer that seems imprecise and unsatisfactory to me. Then I posted it here on MathOverflow for answers of higher quality, and it was also well-received until a user began complaining about its "rapid" cross-posting, claiming that the answer in MSE is correct.

Possibly because of that user's comment, other MathOverflow users rushed to down-vote my question and some also voted to close my post and made it "on-hold".

I asked the answerer in my MSE post to provide a more precise answer but he refused. Also it seems some other high reputation users who have some special interests in power towers, share the same point of view with me because they provided some partial answers that clearly shows they are not sure about the correctness of the first answerer's intuitively written answer.

Was casting close-votes on my question correct or it was just a too quick misjudgment?

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    $\begingroup$ I still counsel patience. Right now this is a question within the purview of Mathematics StackExchange, and you are within your rights there to press for more if you are not satisfied. (It's not at all clear that the other user -- note the singular -- "shares your point of view", but this is not a concern for MO.) $\endgroup$ – Todd Trimble Oct 7 '15 at 13:28
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    $\begingroup$ "I asked the answerer in my MSE post to provide a more precise answer but he refused." This seems an odd way to put it; you received additional information there. $\endgroup$ – user9072 Oct 7 '15 at 13:29
  • $\begingroup$ @quid That was the case when I was writing this post. After that the answerer replied to my comment. However it didn't make his answer any clearer. $\endgroup$ – Amit Sing Mukerjee Oct 7 '15 at 13:58
  • $\begingroup$ @ToddTrimble I think people in MathOverflow think every question that is posted on M.SE. is a silly undergraduate question and don't pay enough attention to it. Maybe it was better to post my question here directly or without mentioning that I posted it in M.SE. In this way people won't vote a question blindly and without enough reflecting on its different aspects. $\endgroup$ – Amit Sing Mukerjee Oct 7 '15 at 14:06
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    $\begingroup$ There was a reply long before you posted this (like more than 12h and you even had replied). A second reply on your follow up (an hour before you posted this) came within like 15 minutes of your commenting and almost an hour before you posted this here. $\endgroup$ – user9072 Oct 7 '15 at 14:31
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    $\begingroup$ Amit, you couldn't be farther than the truth about "every question on M.SE.". Many questions there are quite substantial. And I've asked questions there myself, so I certainly don't look down on it. Nor do I think your question is "silly". I do think that Ivan has given enough hints that a (perhaps strong) undergraduate would be able to work out all the details to completion, and I still do think that as a result this question would not long remain open at MO, no matter what you did (and of course you know that not mentioning cross-posting is a site violation). $\endgroup$ – Todd Trimble Oct 7 '15 at 14:33
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    $\begingroup$ In a post on meta (unfortunately I didn't keep the link), user Stefan Kohl explained, whenever a question has a vote to close, he also votes to close. That's how questions get closed on MO. $\endgroup$ – Todd Leason Oct 7 '15 at 15:57
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    $\begingroup$ @ToddLeason this is not at all what he said in the comment you ought to mean (by the way it got self-deleted some time ago, but this is besides the point). $\endgroup$ – user9072 Oct 7 '15 at 16:09
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    $\begingroup$ To put this question out of its misery, I have posted a solution at M.SE. $\endgroup$ – Todd Trimble Oct 7 '15 at 16:26
  • $\begingroup$ @ToddTrimble I think the "answer box" is for posting "answers" not "hints". What you call an answer is nothing but an ambiguous comment trivially observable from definitions. I doubt those who voted to close my question (that many people here and in M.SE. found interesting) bothered themselves to reflect on it. I really see no similarity between your and Ivan's answer. Though, I understand if you mentioned that statements about the similarity of two answers in the beginning of your answer, for the sake of politeness and for not offending Ivan. $\endgroup$ – Amit Sing Mukerjee Oct 7 '15 at 17:08
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    $\begingroup$ Could you please take your ranting about an answer or a hint or whatever on a site other than this one elsewhere. $\endgroup$ – user9072 Oct 7 '15 at 17:15
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    $\begingroup$ I think Daniel Soltesz has already addressed this: likely the user Ivan figured a short answer with the essential idea would be enough for someone with sufficient experience, which is probably what he took you to be. I don't at all agree that his observations were trivial, and I'm afraid that I am going to have to insist that I took his answer as a template for my own more detailed answer, and I wasn't merely being polite. I did have to supply an extra argument for Case 2, but it was nothing. And now, with quid, I'm going to ask that this discussion end. Glad I could help. $\endgroup$ – Todd Trimble Oct 7 '15 at 17:17
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    $\begingroup$ Yes, this is the wrong site to argue about something happening at M.SE, as quid explained. If I were you, I would think carefully before saying another word on this topic here. $$ $$ If you have a question about some other topic that pertains to the operation of MathOverflow, then you may ask about it here. If you have a new question that you think is appropriate for MO, you may ask about it there. But please choose your words wisely. I think I've about gotten my fill recently of people who go on and on past the limits of welcome. $\endgroup$ – Todd Trimble Oct 7 '15 at 17:38
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    $\begingroup$ I agree with everything @ToddTrimble has said in this comment thread. As a professional mathematical analyst (of some kind), I have read Ivan Neretin's sketch of a solution on MSE, and it is far more than "an ambiguous comment trivially observable from definitions" - it is exactly the kind of outline that analysts give each other when they trust each other to fill in the routine details. I appreciate that Ivan's answer may have been unsatisfactory for you, in which case the thing to do would have been to engage with him in MSE comments or chat to seek clarification, rather than complain here. $\endgroup$ – Yemon Choi Oct 7 '15 at 18:41
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    $\begingroup$ Amit, you are very welcome to complain in public about any of my actions on this site. But please let me know about your complaint, using @ or sending an email using the info on my profile. $\endgroup$ – user25199 Oct 8 '15 at 8:31
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The close votes on your question are correct.

You write it correctly that the answer on MSE seems imprecise and unsatisfactory for you. But it looks like it is actually quite precise and satisfactory for the MO community.

From the discussion at MSE it seems that you need a perfectly precise proof with all the epsilons and stuff. This is quite surprising as the question seems advanced enough that one expects that one shouldn't go 100% into details when presenting the proof.

Otherwise it is a cool question, I like it. But your behavior is not suitable for MO, so please take very seriously the comments of senior users and moderators here and next time wait a bit more on MSE before crossposting. (The non existence of such pairs in the range $(e^{-1},\infty)$ can also be seen by the monotoniticy of the infinite power tower and by the way it is convergent/not convergent in this range. But the answer you got at MSE is better (and more detailed) than this, as it can be applied to every positive number.)

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