On the Littlewood Problem Modulo a Prime

Abstract

Abstract. Let p be a prime, and let f : ℤ/p ℤ →ℝ be a function with . Then Minx ∈ℤ/pℤ |f(x)| = O(log p)−1/3+∈. 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 A ⊆ℤ/pℤ is a set of cardinality ⌊p/2⌋, then ∑r |1^A(r)| ≫ (log p)1/3−∈. 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 A ⊆ℤ/pℤ is a set of size ⌊p/2⌋, then there is some x ∈ℤ/pℤ such that‖A ∩ (A + x)| − p/4| = o(p).