Untitled

Abstract

We study pairs of subsets A, B of a compact abelian group G where the sumset A + B := {a + b : a ∈ A, b ∈ B} is small. Let m and m * be Haar measure and inner Haar measure on G, respectively. Given ε > 0, we classify all pairs A, B of Haar measurable subsets of G satisfying m(A), m(B) > ε and m * (A + B) ≤ m(A) + m(B) + δ where δ = δ (ε) is small. We also study the case where the δ-popular sumset A + δ B := {t ∈ G : m(A ∩ (t − B)) > δ } is small. We prove that for all ε > 0, there is a δ > 0 such that if A and B are subsets of a compact abelian group G having m(A), m(B) > ε and m(A + δ B) ≤ m(A) + m(B) + δ , then there are sets S, T ⊆ G such that m(A S) + m(B T) < ε and m(S + T) ≤ m(S) + m(T). Appealing to known results, the latter inequality yields strong structural information on S and T , and therefore on A and B.