Need help with references on the status of a "Littlewood Problem"

Question

The "Littlewood Problem" in the title asks for a characterization of finite sequences

n₁< ...< n_k of integers such that z^n₁+z^n₂+...+z^n_k≠0 for any complex number z of unit modulus.

Does anybody know about the current status of this problem?

Some Background

1)I came to know this Littlewood Problem through the paper of Casazza & Kalton, http://www.jstor.org/pss/2699467.

2)For k=2,3,4, by some simple geometric argument, a complete characterization can be easily obtained. I wonder if such a result has already appeared in the literature.

3)Furthermore, I wonder if at least for the case of k=5, (or indeed, similarly for any k),the following is true? And if it is, whether it is in the literature somewhere.

Suppose that for some complex number z of unit modulus and some integers n₁< ...< n₅,

z^n₁+z^n₂+...+z^n₅=0

then either z^n₁,z^n₂,..,z^n₅ are evenly distributed on the unite circle (i.e., they look like the 5th roots of unit after a certain rotation is applied to each) or three pounts among z^n₁,z^n₂,..,z^n₅ are evenly distributed on the unite circle.

Answer 1

For every $\alpha$, $\pi/3\le\alpha\le\pi$, there is a $\beta$ such that $1+e(\alpha)+e(-\alpha)+e(\beta)+e(-\beta)=0$, where I write $e(x)$ for $e^{2\pi ix}$. Moreover, $\beta$ depends continuously on $\alpha$, so there will be infinitely many $\alpha$ such that $\beta/\alpha$ is rational, say, $\beta/\alpha=m/n$. Let $z=e(\alpha/n)$. Then $z^0+z^n+z^{-n}+z^m+z^{-m}=0$, and in general these numbers will not be vertices of a regular pentagon, nor include vertices of an equilateral triangle. This is in (negative) answer to part 3) of the question, case $k=5$, and surely the answer is negative as well for any $k\ge5$.