For any locally compact Abelian group, the Hausdorff-Young inequality
states that the Fourier transform maps $L^p$ to $L^q$,
where the two exponents are conjugate and $p \in [1,2]$. For Euclidean space,
the optimal constant in the inequality was found Babenko for $q$ 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 $1<p<2$: (i) If a function $f$ nearly
achieves the optimal constant in the inequality, then $f$ 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.