I have just uploaded to the arXiv the paper “An inverse theorem for an inequality of Kneser“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kneser discussed previously in this blog, namely that $\displaystyle \mu(A+B) \geq \min(\mu(A)+\mu(B), 1) \ \ \ \ \ (1)$

whenever ${A,B}$ are compact non-empty subsets of a compact connected additive group ${G}$ with probability Haar measure ${\mu}$.  (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime $\displaystyle \mu(A), \mu(B), 1- \mu(A)-\mu(B) > 0. \ \ \ \ \ (2)$

The connectedness of ${G}$ is essential, otherwise one could form counterexamples involving proper subgroups of ${G}$ of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as $\displaystyle \mu( (A_1 \cup A_2) + B) + \mu( (A_1 \cap A_2) + B)$ $\displaystyle \leq \mu(A_1+B) + \mu(A_2+B) \ \ \ \ \ (3)$

which in turn easily follows from the identity ${(A_1 \cup A_2) + B = (A_1+B) \cup (A_2+B)}$ and the inclusion ${(A_1 \cap A_2) + B \subset (A_1 +B) \cap (A_2+B)}$, combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking ${G}$ to be the unit circle ${G = {\bf R}/{\bf Z}}$ and ${A,B}$ to be arcs in that circle (obeying (2)). A bit more generally, if ${G}$ is an arbitrary connected compact abelian group and ${\xi: G \rightarrow {\bf R}/{\bf Z}}$ is a non-trivial character (i.e., a continuous homomorphism), then ${\xi}$ must be surjective (as ${{\bf R}/{\bf Z}}$ has no non-trivial connected subgroups), and one can take ${A = \xi^{-1}(I)}$ and ${B = \xi^{-1}(J)}$ for some arcs ${I,J}$ in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then ${A,B}$ must be close (in measure) to an example of the above form ${A = \xi^{-1}(I), B = \xi^{-1}(J)}$. Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset ${A+B}$ is replaced by the partial sumset ${A +_\varepsilon B}$ consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call ${(A,B)}$ a critical pair if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in ${A,B}$, and invariant with respect to translation of either ${A}$ or ${B}$. Furthermore, from the submodularity inequality (3), one can show that if ${(A_1,B)}$ and ${(A_2,B)}$ are critical pairs (with ${\mu(A_1 \cap A_2)}$ and ${1 - \mu(A_1 \cup A_2) - \mu(B)}$ positive), then ${(A_1 \cap A_2,B)}$ and ${(A_1 \cup A_2, B)}$ are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when ${A,B}$ come from arcs of a circle.) Similarly, from associativity ${(A+B)+C = A+(B+C)}$, one can show that if ${(A,B)}$ and ${(A+B,C)}$ are critical pairs, then so are ${(B,C)}$ and ${(A,B+C)}$.

One can combine these closure properties to obtain further ones. For instance, suppose ${A,B}$ is such that ${\mu(A+B) 0}$. Then (cheating a little bit), one can show that ${(A+B,C)}$ is also a critical pair, basically because ${A+B}$ is the union of the ${A+b}$, ${b \in B}$, the ${(A+b,C)}$ are all critical pairs, and the ${A+b}$ all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate ${A+B}$ by the union of finitely many of the ${A+b}$, then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair ${(A,B)}$ and end up with a small set ${C}$ such that ${(A,C)}$ and ${(kC,C)}$ are also critical pairs for all ${1 \leq k \leq 10^4}$ (say), where ${kC}$ is the ${k}$-fold sumset of ${C}$. (Intuitively, if ${A,B}$ are thought of as secretly coming from the pullback of arcs ${I,J}$ by some character ${\xi}$, then ${C}$ should be the pullback of a much shorter arc by the same character.) In particular, ${C}$ exhibits linear growth, in that ${\mu(kC) = k\mu(C)}$ for all ${1 \leq k \leq 10^4}$. One can now use standard technology from inverse sumset theory to show first that ${C}$ has a very large Fourier coefficient (and thus is biased with respect to some character ${\xi}$), and secondly that ${C}$ is in fact almost of the form ${C = \xi^{-1}(K)}$ for some arc ${K}$, from which it is not difficult to conclude similar statements for ${A}$ and ${B}$ and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)

[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman.  Thanks to John Griesmer for pointing out the error.]