An inverse theorem for an inequality of Kneser

TL;DR

This paper proves an inverse theorem characterizing near-equality cases in Kneser's inequality for compact abelian groups, showing that such sets are close to structured examples involving homomorphisms to the circle.

Contribution

It establishes a new inverse theorem for Kneser's inequality, providing a structural description of sets nearly attaining the inequality's bound in compact abelian groups.

Findings

Near-equality in Kneser's inequality implies sets are close to structured examples.

The inverse theorem applies to both exact and approximate sumset cases.

The results have applications to patterns in multiplicative functions.

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 $$ \mu_G(A + B) \geq \min( \mu_G(A)+\mu_G(B), 1 ) $$ 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 $\mu_G(A)+\mu_G(B) \geq 1$. Another way in which equality can be obtained is when $A = \phi^{-1}(I), B = \phi^{-1}(J)$ for some continuous surjective homomorphism $\phi: G \to {\bf R}/{\bf Z}$ and compact arcs $I,J \subset {\bf R}/{\bf Z}$. 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 $A +_\varepsilon B :=\{ 1_A * 1_B \geq \varepsilon \}$ for some small $\varepsilon >0$. In a subsequent paper with Joni Ter\"av\"ainen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.