Let $n$ be a large natural number, and let $z\_1, \ldots, z\_{10}$ be (say) ten $n^{th}$ roots of unity: $z\_1^n = \ldots = z\_{10}^n = 1$. Suppose that the sum $S = z\_1+\ldots+z\_{10}$ is non-zero. How small can $|S|$ be?


$S$ is an algebraic integer in the cyclotomic field of order $n$, so the product of all its Galois conjugates has to be a non-zero rational integer. Using the utterly crude estimate that the magnitude of a non-zero rational integer is at least one, this gives an exponential lower bound on $S$. On the other hand, standard probabilistic heuristics suggest that there should be a polynomial lower bound, such as $n^{-100}$, for $|S|$. (Certainly a volume packing argument shows that one can make $S$ as small as, say, $O(n^{-5/2})$, though it is unclear to me whether this should be close to the true bound.) Is such a bound known? Presumably one needs some algebraic number theoretic methods to attack this problem, but the only techniques I know of go through Galois theory and thus give exponentially poor bounds.


Of course, there is nothing special about the number $10$ here; one can phrase the question for any other fixed sum of roots, though the question degenerates when there are four or fewer roots to sum.



How small can a sum of a few roots of unity be?
In this paper they talk about this problem for 5 instead of 10 roots.


<http://www.jstor.org/stable/2323469>


EDIT: In view of Todd Trimble's comment, here's a summary of what's in the paper. 


Let $f(k,N)$ be the least absolute value of a nonzero sum of $k$ (not necessarily distinct) $N$-th roots of unity. Then 


$f(2,N)$ is asymptotic to $cN^{-1}$, where $c$ is $2\pi$ for even $N$, $\pi$ for odd $N$, 


$f(3,N)$ is asymptotic to $cN^{-1}$, where $c$ is $2\pi\sqrt3$ for $N$ divisible by 3, $2\pi\sqrt3/3$ otherwise, 


$f(4,N)$ is asymptotic to $cN^{-2}$, where $c$ is $4\pi^2$ for even $N$, $\pi^2$ for $N$ odd, 


$f(k,N)>k^{-N}$ for all $k,N$, 


$f(2s,N)<c\_sN^{-s}$ for $N$ even and $s\le10$, 


$f(k,N)<c\_kN^{-[\sqrt{k-6}]-1}$ for $N$ even and $k>5$, and 


If $N$ is twice a prime, and $k<N/2$, then there exists $k'<2k$ such that $f(k',N)\le2k2^{k/2}\sqrt{k!}N^{-k/2}$. 


The only result in the paper for 5 roots of unity is (the trivial) $f(5,N)>5^{-N}$, but it is suggested that maybe $f(5,N)>cN^{-d}$ for some $d$, $2\le d\le3$, and some $c>0$. 


