  Suppose that f is a function from Z_p -> [0,1] (Z_p is my notation for the
integers mod p, not the p-adics), and suppose that a_1,...,a_k are some places
in Z_p. In some additive number theory applications it would be nice to perturb
f slightly so that Fourier transform f^ vanishes at a_1,...,a_k, while additive
properties are left intact. In the present paper, we show that even if we are
unsuccessful in this, we can at least say something interesting by using the
principle of the separating hyperplane, a basic ingredient in linear
programming duality.
