On the Littlewood problem modulo a prime

Let p be a prime, and let f : Z/pZ -> R be a function with average value 0 and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is no more than O(log p)^{-1/3 + eps}. One should think of f as being ``approximately continuous''; our result is then an ``approximate intermediate value theorem''. As an immediate consequence we show that if B in Z/pZ is a set of cardinality (p-1)/2 then the algebra norm ||1_B||_A is >> (log p)^{1/3 - eps}. This gives a result on a ``mod p'' analogue of Littlewood's well-known problem concerning the smallest possible L^1-norm of the Fourier transform of a set of n integers. Another application is to answer a question of Gowers. If B in Z/pZ is a set of size (p-1)/2 then there is some x in Z/pZ such that the intersection of B with B + x has cardinality within o(p) of p/4.