Optimally small sumsets in finite abelian groups

Shalom Eliahou; Michel Kervaire; Alain Plagne

doi:10.1016/S0022-314X(03)00060-X

Optimally small sumsets in finite abelian groups

Shalom Eliahou
, Michel Kervaire
, Alain Plagne

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be a finite abelian group of order g. We determine, for all 1 ≤r,s≤g, the minimal size μ_G(r,s) = min A + B of sumsets A + B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.

Original language	English
Pages (from-to)	338-348
Number of pages	11
Journal	Journal of Number Theory
Volume	101
Issue number	2
DOIs	https://doi.org/10.1016/S0022-314X(03)00060-X
Publication status	Published - 1 Aug 2003

Keywords

Additive number theory
Cauchy-Davenport theorem
Initial segment
Kneser theorem
Sumset

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10.1016/S0022-314X(03)00060-X

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title = "Optimally small sumsets in finite abelian groups",

abstract = "Let G be a finite abelian group of order g. We determine, for all 1 ≤r,s≤g, the minimal size μG(r,s) = min A + B of sumsets A + B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction.",

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AU - Kervaire, Michel

AU - Plagne, Alain

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N2 - Let G be a finite abelian group of order g. We determine, for all 1 ≤r,s≤g, the minimal size μG(r,s) = min A + B of sumsets A + B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction.

AB - Let G be a finite abelian group of order g. We determine, for all 1 ≤r,s≤g, the minimal size μG(r,s) = min A + B of sumsets A + B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction.

KW - Additive number theory

KW - Cauchy-Davenport theorem

KW - Initial segment

KW - Kneser theorem

KW - Sumset

U2 - 10.1016/S0022-314X(03)00060-X

DO - 10.1016/S0022-314X(03)00060-X

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VL - 101

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EP - 348

JO - Journal of Number Theory

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ER -

Optimally small sumsets in finite abelian groups

Abstract

Keywords

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