Michael Christ University of California, Berkeley (UC Berkeley) Mathematics For any locally compact Abelian group, the Hausdorff-Young inequality states that the Fourier transform maps LpLp to LqLq, where the two exponents are conjugate and p∈[1,2]p∈[1,2]. For Euclidean space, the optimal constant in the inequality was found Babenko for qq an even integer, and by Beckner for general exponents. Lieb showed that all extremizers are Gaussian functions. This is a uniqueness theorem; these Gaussians form the orbit of a single function under the group of symmetries of the inequality. We establish a stabler form of uniqueness for $1achieves the optimal constant in the inequality, then ff must be close in norm to a Gaussian. (ii) There is a quantitative bound involving the square of the distance to the nearest Gaussian. The qualitative form (i) can be equivalently formulated as a precompactness theorem in the style of the calculus of variations. Form (ii) is a strengthening of the inequality. The proof relies on ingredients taken from from additive combinatorics. Central to the reasoning are arithmetic progressions of arbitrarily high rank.