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**Critical pairs in Abelian groups and Kemperman’s structure theorem.**
*(English)*
Zbl 1157.11040

Suppose that \(G\) is an abelian group. If \(G\) is torsion free then it is easy to see that any pair of subsets \((A,B)\) satisfies \(|A+B| \geq |A| + |B| -1\) where \(A+B:=\{a+b: a \in A, b \in B\}\). Kemperman’s theorem classifies those pairs \((A,B)\) with \(|A+B| \leq |A| + |B| -1\) in any abelian group.

The statement of the theorem is necessarily rather involved because the extremal sets can be quite complicated and one of the objectives of the paper under review is to provide a readable and motivated exposition of Kemperman’s theorem.

The author has placed considerable effort into giving a well motivated account including Kneser’s theorem and the Kemperman-Scherk theorem. As is pointed out the name for this last theorem is not standard but the statement is sufficiently easy that we may include it. We define \[ \mu(A,B):=\inf_{x \in A+B}{1_A \ast 1_B(x)} \] and then the Kemperman-Scherk theorem asserts that \[ |A+B| \geq |A| + |B| - \mu(A,B). \] As well as being an excellent expository work the paper also proves a new theorem which the author describes as in some sense dual to Kemperman’s classification. This is applied to improve some results of Deshouillers, Hamidoune, Hennecart, and Plagne and rounds out the paper nicely.

The statement of the theorem is necessarily rather involved because the extremal sets can be quite complicated and one of the objectives of the paper under review is to provide a readable and motivated exposition of Kemperman’s theorem.

The author has placed considerable effort into giving a well motivated account including Kneser’s theorem and the Kemperman-Scherk theorem. As is pointed out the name for this last theorem is not standard but the statement is sufficiently easy that we may include it. We define \[ \mu(A,B):=\inf_{x \in A+B}{1_A \ast 1_B(x)} \] and then the Kemperman-Scherk theorem asserts that \[ |A+B| \geq |A| + |B| - \mu(A,B). \] As well as being an excellent expository work the paper also proves a new theorem which the author describes as in some sense dual to Kemperman’s classification. This is applied to improve some results of Deshouillers, Hamidoune, Hennecart, and Plagne and rounds out the paper nicely.

Reviewer: Tom Sanders (Cambridge)

### MSC:

11P70 | Inverse problems of additive number theory, including sumsets |

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\textit{V. F. Lev}, Int. J. Number Theory 2, No. 3, 379--396 (2006; Zbl 1157.11040)

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