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If I like your problem, I'll think of it in honest. However, if the question is badly posed or the information is incomplete, I'll vote to close it when in a bad mood and waste several hours of your and my time asking for clarification, pointing out trivial counterexamples, etc., when in a good mood. In both cases, the most likely outcome is that I'll finally switch my attention to something else. ;)
7h

awarded  Good Answer 
21h

comment 
Uniformly small sums of roots of unity
One possibility is to try to do something similar to Spencer's "Six standard deviations suffice" (note that the full sum is 0, so you can restate it in terms of "mean zero" coefficients, say $\pm 1$ for halfsize sets), but that will give only about $6n$, not $n/2$... 
1d

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Uniformly small sums of roots of unity
@Fedor Petrov Ah, yes; I misread the problem quite a bit :). OK, let me think then... 
1d

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Uniformly small sums of roots of unity
Something is fishy: if we just take any pair of "almost opposite" roots of unity, the absolute value of their sum is of order $1/n$, so you can easily get constant instead of $\sqrt n$ for large $n$. 
Jun 18 
comment 
What would you do if you improve your own result that is submitted but not publishied?
Since it is a personal question (what would YOU do...), I'll give a personal answer: Step 1. Check that you do not have a hallucination and the new result is real and foolproof. Step 2. Tell the editors that you want to do a major overhaul of the paper in view of new developments Step 3. Do the overhaul and send the new version to the same journal. The rationale is that normally there is no reason to publish a result if a much better one is available and the only agreement you have with the editors is that you chose their journal to present your work on that topic, and even that is not binding 
Jun 16 
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Boundedness of solutions of a difference equation
@salimmath15 This conjecture need many experts in dynamical system and bifurction to proof it. How can you possibly know that? At this point it is just a simply stated question that has not been answered by anyone you asked to think of it. Be patient and try to avoid estimating the difficulty of an unsolved problem until you see the sixth solution :) 
Jun 12 
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Boundedness of solutions of a difference equation
@Iosif Pinelis The equivalence is (all solutions are bounded)\equiv(\beta=\lambda). The "all" quantifier is a part of the statement on the left, so one exceptional initial value set is not disproving anything... 
Jun 9 
comment 
$BMO$property via a JohnNirenberg type estimate?
Unfortunately, nothing short of (one of many equivalent forms of) the definition itself works. So, in this generality the question is unanswerable. Try to ask exactly what you need and then you may have a chance. 
Jun 6 
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Dose any infinite dimensional subspace contain the range of some one to one bounded linear operator?
First of all, for any $n$, you can find a unit vector $v_n$ such that $Tv_n$ has a nonzero coordinate corresponding to the index $m>n$. Now make a sum $v=\sum_n a_nv_n$ with random $a_n\in (0,n^{2})$. Then $Tv=\sum a_nTv_n$ has infinitely many nonzero coordinates with probability $1$. @Bill Johnson You are right, of course, but my experience with googling key words for a simple question is that you get a lot of highly sophisticated papers for each of which it takes more effort to figure out if the answer is there than to solve the problem yourself :) 
Jun 6 
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Dose any infinite dimensional subspace contain the range of some one to one bounded linear operator?
@paul garrett We can certainly discuss that but I'd like to see the looming closure relieved first. One way to do it is to ask a new question in the form that would provoke such a discussion. Do you want to do it, Paul? 
Jun 5 
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Dose any infinite dimensional subspace contain the range of some one to one bounded linear operator?
No. Take the linear subspace of finitely supported vectors in $\ell^2$. I wonder what is the example of the person who downvoted... 
Jun 4 
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An inequality for two independent identically distributed random vectors in a normed space
@BillJohnson with a few more points and 3.95 I can get an espresso Actually, I would rather redeem them for getting more people who read other posts of mine (and that does not even require $3.95 !). Believe it or not, there are many souls out there who would benefit from learning from you (even if they disagree to you to the extent that they would rather challenge you to a shooting duel at 20 yards than recognize that they owe you something). More seriously, the reputation points are just a numerical quantity reflecting something very real: your professional reputation... 
Jun 3 
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An inequality for two independent identically distributed random vectors in a normed space
No problem whatsoever. I said everything I have to say on this issue long ago (see mathoverflow.net/questions/66162/… ) and it takes way more than 4 years to change my core opinions :). My only request to the people who upvote my answer is to upvote Bill's one as well. He really deserves it, IMHO :). Now I'll take care of some other fish to fry for a while... 
Jun 2 
awarded  Nice Answer 
Jun 2 
revised 
An inequality for two independent identically distributed random vectors in a normed space
added 6 characters in body 
Jun 2 
revised 
An inequality for two independent identically distributed random vectors in a normed space
added 7 characters in body 
Jun 2 
answered  An inequality for two independent identically distributed random vectors in a normed space 
May 20 
awarded  Good Answer 
May 15 
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Modern Mathematical Achievements Accessible to Undergraduates
It is. But read the second sentence in my post; that part took me more than 24 hours, indeed :) 
May 7 
awarded  Good Answer 