[math-fun] another Roots-of-unity puzzle

Dan Asimov <dasimov@earthlink.net> posed the following Roots-of-unity puzzle:

For each n >= 2, find f(n) = the size of the largest subset of the n complex nth roots of unity that does not contain k equally spaced points for any k >= 2.

(I.e., which does not contain a congruent copy of the k complex kth roots of unity for any k >= 2.)

I'll pose a modification -- find the subsets from Dan's puzzle whose root sum has minimum absolute value. Can someone prove or disprove that this minimum absolute value > 0 ? - Scott