7

Let $n$ be an integer, $n\ge2$. Let $m$ be a positive integer, $m\le n$, having no common factor with $n$. How can we select $m$ distinct complex $n$th roots of unity in such a way as to minimise the modulus of their sum?

The condition that $m$ and $n$ be coprime is included because if $m$ and $n$ have a common factor, it is easy to see that we can always find a sum which vanishes.

The obvious answer is that the roots chosen should be spread around the unit circle as evenly as possible. This does not appear to be correct: for example, if $n=9$, $m=4$ and if we write $\alpha=e^{2\pi i/9}$, then $$|1+\alpha^2+\alpha^4+\alpha^6|=2\cos\frac{2\pi}{9}-1=0.5321\ ,$$ while $$|1+\alpha+\alpha^4+\alpha^6|=2\cos\frac{\pi}{9}-2\cos\frac{2\pi}{9} =0.3473\ .$$

David
  • 82,662
  • Is something known about the distribution of euclidean norm of the centroid of $m$-uples of $n$-th roots of unity? I am thinking that the minimum may be very difficult to find, but we may have something like "there is a choice that grants a distance less than $\frac{2}{\sqrt{m}}$" through the probabilistic method. – Jack D'Aurizio Aug 11 '14 at 04:56
  • We can also use something like (assuming $p\mid m$) choosing $p$ roots every time, minimizing their sum, then choosing other $p$ roots among the remaining one, minimizing their sum... – Jack D'Aurizio Aug 11 '14 at 05:01
  • 2
    There is the nice $m=5,n=12$ where $5=3+2$, and you choose a trio with sum=0, and a pair with sum=0. – Empy2 Aug 11 '14 at 05:46
  • @Michael good observation! That suggests that it may be possible to get a zero sum whenever $m$ is a "suitable" sum of small coprime factors of $n$. – David Aug 11 '14 at 06:09
  • 2
    All the cases with $m\ge5$ are hard (the cases with $m\le4$ are easy). You can see my paper, How Small Can a Sum of Roots of Unity Be?, The American Mathematical Monthly, Vol. 93, No. 6 (Jun. - Jul., 1986), pp. 457-459. See also http://mathoverflow.net/questions/46068/how-small-can-a-sum-of-a-few-roots-of-unity-be – Gerry Myerson Aug 11 '14 at 06:12
  • Thanks @Gerry, will look it up. – David Aug 12 '14 at 01:26

0 Answers0