fa.functional analysis - Need help with references on the status of a "Littlewood Problem"

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?

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.