Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series expansion of $f$, what are possible values of $k$? Any information about what $k$ can or cannot be depending on $p$ and $n$ will be helpful.
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$\begingroup$ This would seem to be closely related to the question of how small a (nonzero) sum of roots of unity can be, which is the topic of another question on this websute. $\endgroup$– Gerry MyersonCommented Jul 29, 2018 at 7:17
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$\begingroup$ @GerryMyerson I have seen that post. Can you give more details on how the two questions are connected? $\endgroup$– DavitSCommented Jul 29, 2018 at 7:23
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2$\begingroup$ After choosing the standard branch for log (which will cause no problem since p is odd), and taking derivatives, this question reduces to one about equal sums of powers, specifically the maximal k s.t. $\sum a_i^k=\sum b_i^k$ where $x_i=exp(2\pi \sqrt{-1} a_i/p)$, $a_i\in[-p/2,p/2]$, and similarly for $b_i$. $\endgroup$– Dror SpeiserCommented Jul 29, 2018 at 10:27
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1$\begingroup$ Are you looking at mathoverflow.net/questions/46068/…, Random? Keep in mind the comment here from @Dror, and look at my comment back there on 15 Nov 2010 at 3:35, also my comment at 20:51 on Denis' answer, also Aaron's answer. $\endgroup$– Gerry MyersonCommented Jul 29, 2018 at 11:03
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