On the Littlewood problem modulo a prime

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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 (L¹ norm of the Fourier transform). Then f(x) is small for some x, specifically minₓ |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 ||1B||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 L¹-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.