Here the pdf version: link.
In the following, I shall use to denote both the Lebesgue measure of , when a subset of , or the cardinality of set . This shouldn’t cause any confusion, and help highlight the parallel with the continuous case.
For the sake of completeness, we remind the reader that the Minkowski sum of two sets is defined as
I’ve been shamefully sketchy in the previous post about Christ’s work on near extremizers, and in particular I haven’t addressed properly one of the most important ideas in his work: exploiting the hidden additive structure of the inequalities. I plan to do that in this post and a following one, in which I’ll sketch his proof of the sharpened Riesz-Sobolev inequality.
In that paper, one is interested in proving that triplets of sets that nearly realize equality in Riesz-Sobolev inequality
must be close to the extremizers of the inequality, which are ellipsoids (check this previous post for details and notation). In case , ellipsoids are just intervals, and one wants to prove there exist intervals s.t. are very small.
Christ devised a tool that can be used to prove that a set on the line must nearly coincide with an interval. It’s the following
Proposition 1 (Christ, [ChRS2]) , (continuum Freiman’s theorem) Let be a measurable set with finite measure . If
then there exists an interval s.t.  and
Thus if one can exploit the near equality to spot some additive structure, one has a chance to prove the sets must nearly coincide with intervals. It turns out that there actually is additive structure concealed in the Riesz-Sobolev inequality: consider the superlevel sets
then one can prove that
If one can control the measure of the set on the right by for some specific value of , then Proposition 1 can be applied, and will nearly coincide with an interval. Then one has to prove this fact extends to , but that’s what the proof in [ChRS] is about and I will address it in the following post, as said.
Anyway, the result in Prop. 1, despite being stated in a continuum setting, is purely combinatoric. It follows – by a limiting argument – from a big result in additive combinatorics: Freiman’s theorem.
The aim of this post is to show how Prop. 1 follows from Freiman’s theorem, and to prove Freiman’s theorem with additive combinatorial techniques. It isn’t necessary at all in order to appreciate the results in [ChRS], but I though it was nice anyway. I haven’t stated the theorem yet though, so here it is:
Theorem 2 (Freiman’s theorem) Let be finite and such that
Then there exists an arithmetic progression s.t. , whose length is .
The proof isn’t extremely hard but neither it’s trivial. It relies on a few lemmas, and it is fully contained in section 2. Section 1 contains instead the limiting procedure mentioned above that allows to deduce Proposition 1 from Freiman’s theorem.
Remark 1 Notice that Proposition 1 is essentially a result for the near-extremizers of Brunn-Minkowski’s inequality in , which states that . Indeed the extremizers for B-M are convex sets, which in dimension 1 means the intervals. Thus Prop 1 is saying that if isn’t much larger than , then is close to being an extremizer, i.e. an interval. One can actually prove that for two sets , if one has
then . A proof can be found in [ChRS]. It is in this sense that the result in [ChBM] for Brunn-Minkowski was used to prove the result in [ChRS] for Riesz-Sobolev, which was then used for Young’s and thus for Hausdorff-Young, as mentioned in the previous post.