  In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1]
has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds
for quasinorms 1/2-\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not
too ``small'', then there are lots of triples m,m+d,m+2d such that
f(m)f(m+d)f(m+2d) > 0. If f is the indicator function for some set S, then this
would be saying that the set has many three-term arithmetic progressions. In
principle this theorem can be applied to sets having very low density, where
|S| is around p^{n(1-c)} for some small c > 0.
