A constrained optimization problem for the Fourier transform

Among functions f majorized by indicator functions of sets E of measure 1, which functions have maximal Fourier transform in L^p? We investigate the existence of maximizers, using a concentration compactness approach and ingredients from additive combinatorics to establish properties of maximizing sequences. For exponents q sufficiently close to even integers, we exploit variational techniques and combinatorial results to identify all maximizers. This follows from establishing a sharper version of an associated inequality: if the input f, where |f| is less than or equal to the indicator function of a measure 1 set E, has a certain structure, then the Fourier transform of f in L^q is at least a certain quantitative distance from being optimal