I want to prove or find a counterexample of the following proposition:

Let $N$ be a positive integer and $a_1,\dotsc,a_N$ be distinct real numbers. Then it holds that: $$\sum_{1 \leq n, m \leq N} \cos(a_n - a_m) \geq 0.$$

For $N=1,2$ the result is obvious and for $N=3,4$ Wolfram Alpha affirms that the result is positive.

Can someone help me here?

  • 1
    $\begingroup$ @MartinR For $N=2$ the sum is $\cos(a_1-a_1)+\cos(a_1-a_2)+\cos(a_2-a_1)+\cos(a_2-a_2)=2+2\cos(a_1-a_2)$, which is clearly $\geq 0$. $\endgroup$ – Gabriel Feb 21 '18 at 21:31

If we define $z$ as $$ z = \sum_{n=1}^{N}\exp(ia_n) $$ we may notice that $$ z\overline{z} = \|z\|^2 = \sum_{1\leq m,n\leq N}e^{i(a_n-a_m)},\qquad \sum_{1\leq m,n\leq N}\cos(a_n-a_m)=\text{Re}\|z\|^2$$ and $\|z\|^2$ is obviously real and $\geq 0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.