Subsets with a Small Sum II: the Critical Pair Problem
A basic problem raised by Kneser was to describe all the finite subsets A, B of an abelian group G such that | A+B | ≤ | A | + | B | − 1. Let G be an abelian group. The description of the pairsA , B⊂G such that | A+B | = | A | + | B | − 1 < |G | was considered in additive group theory. Vosper (19...
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| Veröffentlicht in: | European journal of combinatorics 2000-02, Vol.21 (2), p.231-239 |
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| container_title | European journal of combinatorics |
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| creator | Ould Hamidoune, Yahya |
| description | A basic problem raised by Kneser was to describe all the finite subsets A, B of an abelian group G such that | A+B | ≤ | A | + | B | − 1. Let G be an abelian group. The description of the pairsA , B⊂G such that | A+B | = | A | + | B | − 1 < |G | was considered in additive group theory. Vosper (1956) solved this problem completely for groups with a prime order. Kempermann’s theory for small sums describes the structure of these pairs, ifA+B is aperiodic or if there exists a uniquely expressible element inA+B. In this paper we study the same question with a fixed subset B satisfying the inequality: for all A such that 1 ≤ | A | |
| doi_str_mv | 10.1006/eujc.1999.0340 |
| format | Article |
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Let G be an abelian group. The description of the pairsA , B⊂G such that | A+B | = | A | + | B | − 1 < |G | was considered in additive group theory. Vosper (1956) solved this problem completely for groups with a prime order. Kempermann’s theory for small sums describes the structure of these pairs, ifA+B is aperiodic or if there exists a uniquely expressible element inA+B. In this paper we study the same question with a fixed subset B satisfying the inequality: for all A such that 1 ≤ | A | <∞, |A+B | ≥min(| G |, | A | + |B | − 1). We obtain a recursive description for the subsets A such that |A+B | ≤ | A | + | B | − 1. As corollary of our description, we obtain the following result which implies some limitations of Kempermann’s theory. Suppose that B is neither a coprogression nor almost periodic and that 2 ≤ |A | ≤ | G | − | B | − 1. If |A+B | = | A | + | B | − 1, then A is periodic and A+B contains no unique expression elements. 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Let G be an abelian group. The description of the pairsA , B⊂G such that | A+B | = | A | + | B | − 1 < |G | was considered in additive group theory. Vosper (1956) solved this problem completely for groups with a prime order. Kempermann’s theory for small sums describes the structure of these pairs, ifA+B is aperiodic or if there exists a uniquely expressible element inA+B. In this paper we study the same question with a fixed subset B satisfying the inequality: for all A such that 1 ≤ | A | <∞, |A+B | ≥min(| G |, | A | + |B | − 1). We obtain a recursive description for the subsets A such that |A+B | ≤ | A | + | B | − 1. As corollary of our description, we obtain the following result which implies some limitations of Kempermann’s theory. Suppose that B is neither a coprogression nor almost periodic and that 2 ≤ |A | ≤ | G | − | B | − 1. If |A+B | = | A | + | B | − 1, then A is periodic and A+B contains no unique expression elements. 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| title | Subsets with a Small Sum II: the Critical Pair Problem |
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