On the Hardy-Littlewood majorant problem for random sets

The Hardy-Littlewood majorant problem asks whether L^p norms of functions on the circle grow if one replaces their Fourier coefficients with their absolute values. This is clear if p is an even integer, but false if p is any other number. One can still ask if the norm grows at most by the degree raised to a small power for any p. We show that this is so with any epsilon power provided the majorizing function is the Dirichlet kernel on a random set, with large probability.