##
**A step beyond Kemperman’s structure theorem.**
*(English)*
Zbl 1213.11179

From the introduction: We extend Kemperman’s structure theorem by completely characterizing those finite subsets \(A\) and \(B\) of an arbitrary abelian group with \(| A + B|= || A| + | B|\).

Let \(G\) be an abelian group, and let \(A\) and \(B\) be nonempty subsets of \(G\). Their sumset is the set of all pairwise sums, i.e., \(A+B = \{a+b\mid a\in A,\;b\in B\}\). For a set \(S\) of subsets of \(G\), define \(d^{\subseteq}(A, S) = \min_{B\in S} {d^{\subseteq}(A,B)}\), where \(d^{\subseteq}(A,B) = | B\backslash A|\), if \(A \subseteq B\), and \(d^{\subseteq}(A,B) = \infty\) otherwise. Hence \(d^{\subseteq}(A, S)\) measures how far away as a subset the set \(A\) is from the sets \(B\in S\).

It is the central problem of inverse additive theory to describe the structure of those pairs of subsets \(A\) and \(B\) with \(| B + A|\) small. Such descriptions often prove useful to other related areas of mathematics – a notable example being the use of Freiman’s Theorem, describing \(A\subseteq\mathbb Z\) with \(| A + A| < c\,| A|\), to give a more quantitative proof of Szemerédi’s Theorem concerning the existence of (4-term) arithmetic progressions in a subset of positive upper density.

One of the classical results of inverse additive theory was the complete recursive description given by J. H. B. Kemperman [Acta Math. 103, 63–88 (1960; Zbl 0108.25704)] of the ‘critical pairs’ in an abelian group, i.e., those finite, nonempty subsets \(A\) and \(B\) such that \(| A + B| < | A| + | B|\). Among other applications – including results in graph theory and zero-sum additive theory – Kemperman’s Structure Theorem (KST) yields the descriptions of those subsets of a locally compact abelian group whose Haar measure of the sumset fails to satisfy the triangle inequality. Other applications may also be found. ...

In this paper, we move one step beyond KST by completing the description of all subsets \(A\) and \(B\) that exactly achieve (rather than fail to achieve) the triangle inequality, namely for which \(| A + B| = | A| + | B|\). Our main result is Theorem 4.1 (for the full details see the paper) which shows that with a few noted exceptions – all but one in the same vein as the original recursive description of KST – then such \(A\) and \(B\) must be large subsets of a critical pair; more specifically, there must exist \(A' \supseteq A\) and \(B'\supseteq B\) such that \(| A' + B'| = | A'| + | B'| -1\), and which contain \(A\) and \(B\) each with at most one hole, i.e., \(| A' \backslash A|\leq 1\) and \(| B' \backslash B|\leq 1\). Thus in the case \(G = \mathbb Z/p\mathbb Z\), with \(p\) prime, Theorem 4.1 generalizes the prime case completed by Hamidoune and Rødseth, and is the corresponding composite extension of KST. Theorem 4.1 and KST will also yield necessary and sufficient conditions for \(| A + B| = | A| + | B|\).

We should remark that Y. O. Hamidoune, O. Serra and G. Zémor very recently [Combinatorica 28, No. 4, 441–467 (2008; Zbl 1192.11071)] established a particular case of Theorem 4.1, under a series of added assumptions, including that \(\gcd(| G|, 6) = 1\), that \(A\) be a generating subset, that the order of every element of \(A\setminus 0\) be at least \(| A| + 1\), and that a few smaller technical assumptions also hold – the hypotheses needed for their result, particularly the assumption on the order of elements, parallel other first-attempt generalizations of additive results, from the prime order case to the more general abelian group setting.

Let \(G\) be an abelian group, and let \(A\) and \(B\) be nonempty subsets of \(G\). Their sumset is the set of all pairwise sums, i.e., \(A+B = \{a+b\mid a\in A,\;b\in B\}\). For a set \(S\) of subsets of \(G\), define \(d^{\subseteq}(A, S) = \min_{B\in S} {d^{\subseteq}(A,B)}\), where \(d^{\subseteq}(A,B) = | B\backslash A|\), if \(A \subseteq B\), and \(d^{\subseteq}(A,B) = \infty\) otherwise. Hence \(d^{\subseteq}(A, S)\) measures how far away as a subset the set \(A\) is from the sets \(B\in S\).

It is the central problem of inverse additive theory to describe the structure of those pairs of subsets \(A\) and \(B\) with \(| B + A|\) small. Such descriptions often prove useful to other related areas of mathematics – a notable example being the use of Freiman’s Theorem, describing \(A\subseteq\mathbb Z\) with \(| A + A| < c\,| A|\), to give a more quantitative proof of Szemerédi’s Theorem concerning the existence of (4-term) arithmetic progressions in a subset of positive upper density.

One of the classical results of inverse additive theory was the complete recursive description given by J. H. B. Kemperman [Acta Math. 103, 63–88 (1960; Zbl 0108.25704)] of the ‘critical pairs’ in an abelian group, i.e., those finite, nonempty subsets \(A\) and \(B\) such that \(| A + B| < | A| + | B|\). Among other applications – including results in graph theory and zero-sum additive theory – Kemperman’s Structure Theorem (KST) yields the descriptions of those subsets of a locally compact abelian group whose Haar measure of the sumset fails to satisfy the triangle inequality. Other applications may also be found. ...

In this paper, we move one step beyond KST by completing the description of all subsets \(A\) and \(B\) that exactly achieve (rather than fail to achieve) the triangle inequality, namely for which \(| A + B| = | A| + | B|\). Our main result is Theorem 4.1 (for the full details see the paper) which shows that with a few noted exceptions – all but one in the same vein as the original recursive description of KST – then such \(A\) and \(B\) must be large subsets of a critical pair; more specifically, there must exist \(A' \supseteq A\) and \(B'\supseteq B\) such that \(| A' + B'| = | A'| + | B'| -1\), and which contain \(A\) and \(B\) each with at most one hole, i.e., \(| A' \backslash A|\leq 1\) and \(| B' \backslash B|\leq 1\). Thus in the case \(G = \mathbb Z/p\mathbb Z\), with \(p\) prime, Theorem 4.1 generalizes the prime case completed by Hamidoune and Rødseth, and is the corresponding composite extension of KST. Theorem 4.1 and KST will also yield necessary and sufficient conditions for \(| A + B| = | A| + | B|\).

We should remark that Y. O. Hamidoune, O. Serra and G. Zémor very recently [Combinatorica 28, No. 4, 441–467 (2008; Zbl 1192.11071)] established a particular case of Theorem 4.1, under a series of added assumptions, including that \(\gcd(| G|, 6) = 1\), that \(A\) be a generating subset, that the order of every element of \(A\setminus 0\) be at least \(| A| + 1\), and that a few smaller technical assumptions also hold – the hypotheses needed for their result, particularly the assumption on the order of elements, parallel other first-attempt generalizations of additive results, from the prime order case to the more general abelian group setting.

Reviewer: Olaf Ninnemann (Uffing am Staffelsee)

### MSC:

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

11B30 | Arithmetic combinatorics; higher degree uniformity |

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\textit{D. J. Grynkiewicz}, Mathematika 55, No. 1--2, 67--114 (2009; Zbl 1213.11179)

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