Optimal Bounds on the Growth of Iterated Sumsets in Abelian Semigroups
[Bornes optimales sur la croissance des sommes itérées d’ensembles dans les semigroupes abéliens]
Eliahou, Shalom1 ; Mazumdar, Eshita2
1 LMPA-ULCO, Calais (France)
2 Ahmedabad University (India)
Annales de l'Institut Fourier, Online first, 19 p.

Nous obtenons des bornes supérieures optimales sur la croissance des sommes itérées hA=A+⋯+A de sous-ensembles finis A de semigroupes abéliens. Plus précisément, nous montrons que les nouvelles bornes supérieures récemment obtenues via le Théorème de Macaulay en algèbre commutative sont les meilleures possibles, autrement dit sont effectivement atteintes par des sous-ensembles appropriés de semigroupes abéliens appropriés. Nos constructions, dans un langage multiplicatif, sont basées sur certains idéaux monomiaux spécifiques dans des algèbres de polynômes et sur leurs déformations en idéaux binomiaux convenables via des bases de Gröbner.

We provide optimal upper bounds on the growth of iterated sumsets hA=A+⋯+A for finite subsets A of abelian semigroups. More precisely, we show that the new upper bounds recently derived from Macaulay’s theorem in commutative algebra are best possible, i.e., are actually reached by suitable subsets of suitable abelian semigroups. Our constructions, in a multiplicative setting, are based on certain specific monomial ideals in polynomial algebras and on their deformation into appropriate binomial ideals via Gröbner bases.

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DOI : 10.5802/aif.3674
Classification : 11P70, 05E40, 11B13, 13P25
Keywords: Additive combinatorics, Plünnecke inequality, Standard graded algebra, Hilbert function, Binomial representation, Lexideal, Gröbner basis
Mots-clés : Combinatoire additive, inégalité de Plünnecke, algèbre graduée standard, fonction de Hilbert, représentation binomiale, lexidéal, base de Gröbner

Eliahou, Shalom 1 ; Mazumdar, Eshita 2

1 LMPA-ULCO, Calais (France)
2 Ahmedabad University (India) 
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Eliahou, Shalom; Mazumdar, Eshita. Optimal Bounds on the Growth of Iterated Sumsets in Abelian Semigroups. Annales de l'Institut Fourier, Online first, 19 p.

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