The Littlewood-Gowers problem
arXiv:math/0605522 · doi:10.1007/s11854-007-0005-1
Abstract
We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - ε} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.
31 pp. Corrected typos. Updated references.