An inverse theorem for an inequality of Kneser

2017/11/12 by Terence Tao, Tao, Terence
Mathematics · #11B30 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1711.04337

openalex publication_date 2017/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 and Raikov) establishes the bound μ_G(A + B) ≥ min( μ_G(A)+μ_G(B), 1 ) whenever A,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), B = ϕ^-1(J) for some continuous surjective homomorphism ϕ: G → \bf R/\bf Z and compact arcs I,J ⊂ \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 +_ε B :=\ 1_A * 1_B ≥ 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.

Citations

• The Structure of Critical Product Sets
• On sets with small sumset in the circle
• Approximate invariance for ergodic actions of amenable groups

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