An Inverse Theorem for an Inequality of Kneser

Abstract

Let G = (G, +) be a compact connected abelian group, and let μ G denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound μ G (A + B) ≥ min(μ G (A) + μ G (B), 1) whenever A and B are compact subsets of G, and A + B:= {a + b: a ∈ A, b ∈ B} denotes the sumset of A and B. Clearly one has equality when μ G (A) + μ G (B) ≥ 1. Another way in which equality can be obtained is when A = φ −1 (I) and B = φ −1 (J) for some continuous surjective homomorphism φ: G → ℝ/ℤ and compact arcs I, J ⊂ ℝ/ℤ. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then A and 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 + ε B:= {1A * 1B ≥ ε} for some small ε > 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.