What are the properties of the set of complex numbers that can be expressed as a finite sum of distinct unit complex numbers? (Unit complex numbers are complex numbers with absolute value 1.)
(Sorry my english is not perfect)
It is trivial that every complex number is expressible as a sum of finitely many unit complex numbers, but as far as I am aware, the sum contains some elements more than once.
Can all complex numbers also be constructed from a finite sum of points on the unit circle, each point only counting once? If not, what does the set of these sums look like? What subsets does it contain? What are some numbers not in the set?