A professor at my university gave me the following problem:
For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$
If $m$ is rational, expressed as $\frac{a}{b}$, then let
$$\sum_{k=1}^be^{\frac{2a\pi ik^3}{b}}=c$$
Then we have that $L$ is equal to $\frac{c}{b}$, so that $L=0$ if and only if $c=0$. However, I don't find it an easy problem to determine what rational numbers give $c=0$. And I'm really unsure how to solve the problem for irrational $m$.
Any ideas?