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Adding local short description: "One of several related theorems regarding the sizes of certain sumsets in abelian groups", overriding Wikidata description "theorem" (Shortdesc helper) |
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If <math> G </math> is an abelian group and <math> C </math> is a subset of <math> G </math>, the group <math> H(C):= \{g\in G : g + C = C\} </math> is the ''stabilizer'' of <math> C </math>.
===Cardinality===
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If <math> A </math> and <math> B </math> are nonempty <math>m</math>-measurable subsets of <math> G </math> satisfying <math> m_*(A + B) < m(A) + m(B) </math>, then the stabilizer <math> H:=H(A+B) </math> is compact and open. Thus <math> A+B </math> is compact and open (and therefore <math> m </math>-measurable), being a union of finitely many cosets of <math> H </math>. Furthermore, <math>m(A+B) = m(A+H) + m(B+H) - m(H). </math>
==Equality in connected compact abelian groups==
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can be found when <math> G </math> is the torus <math> \mathbb T:= \mathbb R/\mathbb Z </math> and <math> A </math> and <math> B </math> are intervals. Satz 2 of Kneser's 1956 article<ref name=Kneser56/> says that all examples of sets satisfying equation ({{EquationNote|1}}) with non-null summands are obvious modifications of these. To be precise: if <math> G </math> is a connected compact abelian group with Haar measure <math> m, </math> <math> A </math> and <math> B </math> are <math> m </math>-measurable subsets of <math> G </math> satisfying <math> m(A)>0, m(B)>0 </math>, and equation ({{EquationNote|1}}), then there is a continuous surjective homomorphism <math> \phi: G \to \mathbb T </math> and there are closed intervals <math> I </math>, <math> J </math> in <math> \mathbb T </math> such that <math>A \subseteq \phi^{-1}(I)</math>, <math>B \subseteq \phi^{-1}(J)</math>, <math>m(A) = m (\phi^{-1}(I))</math>, and <math> m(B) = m(\phi^{-1}(J))</math>.
==Notes==
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