# Christ’s result on near-equality in Riesz-Sobolev inequality

It’s finally time to address one of Christ’s papers I talked about in the previous two blogposts. As mentioned there, I’ve chosen to read the one about the near-equality in the Riesz-Sobolev inequality because it seems the more approachable, while still containing one very interesting idea: exploiting the additive structure lurking behind the inequality via Freiman’s theorem.

1. Elaborate an attack strategy

Everything is in dimension ${d=1}$ and some details of the proof are specific to this dimension and don’t extend to higher dimensions. I’ll stick to Christ’s notation.

Recall that the Riesz-Sobolev inequality is

$\displaystyle \boxed{\left\langle \chi_{A} \ast \chi_{B}, \chi_{C}\right\rangle \leq \left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast}\right\rangle} \ \ \ \ \ (1)$

and its extremizers – which exist under the hypothesis that the sizes are all comparable – are intervals, i.e. the intervals are the only sets that realize equality in (1). See previous post for further details. The aim of paper [ChRS] is to prove that whenever ${\left\langle \chi_{A} \ast \chi_{B}, \chi_{C}\right\rangle}$ is suitably close to ${\left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast}\right\rangle}$ (i.e. we nearly have equality) then the sets ${A,B,C}$ are nearly intervals themselves.

# Freiman’s theorem and compact subsets of the real line with additive structure

In the following, I shall use ${|A|}$ to denote both the Lebesgue measure of ${A}$, when a subset of ${\mathbb{R}}$, or the cardinality of set ${A}$. 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 ${A,B}$ is defined as

$\displaystyle A+B:=\{a+b \,:\, a\in A, b\in B\}.$

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 ${A,B,C \subset \mathbb{R}^d}$ that nearly realize equality in Riesz-Sobolev inequality

$\displaystyle \left\langle \chi_{A} \ast \chi_{B}, \chi_{C}\right\rangle \leq \left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast}\right\rangle$

must be close to the extremizers of the inequality, which are ellipsoids (check this previous post for details and notation). In case ${d=1}$, ellipsoids are just intervals, and one wants to prove there exist intervals ${I,J,K}$ s.t. ${A \Delta I, B\Delta J, C\Delta K}$ 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 ${A\subset \mathbb{R}}$ be a measurable set with finite measure ${>0}$. If

$\displaystyle |A+A|< 3|A|,$

then there exists an interval ${I}$ s.t. ${A\subset I}$ [1] and

$\displaystyle |I| \leq |A+A|-|A|.$

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

$\displaystyle S_{A,B}(t):=\{x \in \mathbb{R} \,:\, \chi_A \ast \chi_B (x) > t\};$

then one can prove that

$\displaystyle S_{A,B} (t) - S_{A,B} (t') \subset S_{A,-A}(t+t' - |B|).$

If one can control the measure of the set on the right by ${|S_{A,B} (t)|}$ for some specific value of ${t=t'}$, then Proposition 1 can be applied, and ${S_{A,B}(t)}$ will nearly coincide with an interval. Then one has to prove this fact extends to ${A,B,C}$, 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 ${3k-3}$ theorem) Let ${A\subset \mathbb{Z}}$ be finite and such that

$\displaystyle |A+A| < 3|A|-3.$

Then there exists an arithmetic progression ${P}$ s.t. ${A\subseteq P}$, whose length is ${|P|\leq |A+A|-|A|+1}$.

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 ${\mathbb{R}^1}$, which states that ${|A+A|\geq |A|+|A|}$. Indeed the extremizers for B-M are convex sets, which in dimension 1 means the intervals. Thus Prop 1 is saying that if ${|A+A|}$ isn’t much larger than ${2|A|}$, then ${A}$ is close to being an extremizer, i.e. an interval. One can actually prove that for two sets ${A,B}$, if one has

$\displaystyle |A+B| \leq |A|+|B|+\min(|A|,|B|)$

then ${\mathrm{diam}(A) \leq |A+B|-|B|}$. 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.