The "Littlewood Problem" in the title asks for a characterization of finite sequences
n1< ...< nk of integers such that zn1+zn2+...+znk≠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,
then either zn1,zn2,..,zn5 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 zn1,zn2,..,zn5 are evenly distributed on the unite circle.