On the Littlewood Problem modulo a Prime
نویسنده
چکیده
Let p be a prime, and let f : Z/pZ → R be a function with Ef = 0 and ‖f̂‖1 6 1. Then min x∈Z/pZ |f(x)| = O(log p). 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 |1̂A(r)| ≫ (log p). This gives a result on a “mod p” analogue of Littlewood’s well-known problem concerning the smallest possible L-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).
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