An Inverse Theorem for an Inequality of Kneser

Abstract

Let G = (G, +) be a compact connected abelian group, and let μG denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound μG(A + B) ≥ min(μG(A) + μG(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 μG(A) + μG(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 +εB:= {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.

Locations

• Proceedings of the Steklov Institute of Mathematics - View - PDF
• arXiv (Cornell University) - View - PDF
• Proceedings of the Steklov Institute of Mathematics - View - PDF
• arXiv (Cornell University) - View - PDF
• Proceedings of the Steklov Institute of Mathematics - View - PDF
• arXiv (Cornell University) - View - PDF