On the Hardy-Littlewood majorant problem

Let Lambda subset of or equal to {1,...,N} and let {a(n)}(nis an element ofLambda) be a sequence with \a(n)\ less than or equal to 1 for all n. It is easy to see that parallel toSigma(nis an element ofLambda) a(n)e(ntheta)parallel to(p) less than or equal to parallel toSigma(nis an element ofLambda) e(ntheta)parallel to(p) for every even integer p. We give an example which shows that this statement can fall rather dramatically when p is not an even integer. This answers in the negative a question known as the Hardy-Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.