An inverse theorem for an inequality of Kneser

Abstract

Let $G=(G,+)$ be a compact connected abelian group, and let ${\mu }_G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath and Raikov) establishes the bound $${\mu }_G(A+B)\ge \min ({\mu }_G(A)+{\mu }_G(B),1)$$ whenever $A,B$ are compact subsets of $G$, and $A+B:=\{a+b:a\in A,b\in B\}$ denotes the sumset of $A$ and $B$. Clearly one has equality when ${\mu }_G(A)+{\mu }_G(B)\ge 1$. Another way in which equality can be obtained is when $A={\varphi }^{-1}(I),B={\varphi }^{-1}(J)$ for some continuous surjective homomorphism $\varphi :G\to \mathbf{R}/\mathbf{Z}$ and compact arcs $I,J\subset \mathbf{R}/\mathbf{Z}$. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then $A,B$ are close to one of the above examples. We also give a more "robust" form of this theorem in which the sumset $A+B$ is replaced by the partial sumset $A{+}_{\epsilon }B:=\{1_A\ast 1_B\ge \epsilon \}$ for some small $\epsilon >0$. In a subsequent paper with Joni Teräväinen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.

30 pages, no figures. To appear, Proceedings of the Steklov Institute of Mathematics. A gap in the proof of Theorem 4.6 has been repaired

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• Approximate invariance for ergodic actions of amenable groups
• On sets with small sumset in the circle