SCHWEITZER 2004 Problem 8

Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$ , any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$ .