  The Hausdorff-Young inequality for Euclidean space, in its sharp form due to
Beckner, gives an upper bound for the Fourier transform in terms of Lebesgue
space norms, with an optimal constant. The extremizers have been identified by
Lieb to be the Gaussians. We establish an improved upper bound, for functions
that nearly extremize the inequality, with a negative second term roughly
proportional to the square of the distance to the set of extremizers. One
formulation of this term comes with its own sharp constant.
  The main step is to show that any extremizing sequence is precompact, modulo
the action of the group of natural symmetries of the inequality. This step
relies on inverse theorems of additive combinatorial nature.
