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Vincent Beck; Cédric Lecouvey
Additive combinatorics methods in associative algebras
Confluentes Mathematici, 9 no. 1 (2017), p. 3-27, doi: 10.5802/cml.34
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Class. Math.: 11P70, 20D60
Mots clés: Additive combinatorics, group algebras, Kneser Theorem, associative algebras, monoids

Résumé - Abstract

We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser’s and Hamidoune’s theorems on sumsets and Tao’s theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser’s theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras. Finally, we give lower bounds for the Minkowski product of two subsets in finite monoids by using their associated monoid algebras.

Bibliographie

[1] Christine Bachoc, Oriol Serra & Gilles Zémor, “Revisiting Kneser’s Theorem for Field Extensions”, https://arxiv.org/abs/1510.01354, 2015
[2] Nicolas Bourbaki, Algèbre, Springer Science & Business Media, 2007
[3] George T Diderrich, “On Kneser’s addition theorem in groups”, Proceedings of the American Mathematical Society 38 (1973) no. 3, p. 443-451
[4] Shalom Eliahou & Cédric Lecouvey, “On linear versions of some addition theorems”, Linear and multilinear algebra 57 (2009) no. 8, p. 759-775 Article
[5] William Fulton, Algebraic curves, The Benjamin/Cummings Publishing Company, Inc., 1969
[6] David J Grynkiewicz, “Structural Additive Theory, Dev. Math” 2013
[7] Yahya Ould Hamidoune, “On the connectivity of Cayley digraphs”, European Journal of Combinatorics 5 (1984) no. 4, p. 309-312 Article
[8] Xiang-dong Hou, “On a vector space analogue of Kneser’s theorem”, Linear Algebra and its Applications 426 (2007) no. 1, p. 214-227 Article
[9] Xiang-Dong Hou, Ka Hin Leung & Qing Xiang, “A generalization of an addition theorem of Kneser”, Journal of Number Theory 97 (2002) no. 1, p. 1-9 Article
[10] Florian Kainrath, “On local half-factorial orders”, Arithmetical properties of commutative rings and monoids 241 (2005), p. 316-324 Article
[11] Cédric Lecouvey, “Plünnecke and Kneser type theorems for dimension estimates”, Combinatorica 34 (2014) no. 3, p. 331-358 Article
[12] Diego Mirandola & Gilles Zémor, “Critical pairs for the product singleton bound”, IEEE Transactions on Information Theory 61 (2015) no. 9, p. 4928-4937 Article
[13] Melvyn B Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets 165, Springer, 1996
[14] John E Olson, “On the sum of two sets in a group”, Journal of Number Theory 18 (1984) no. 1, p. 110-120 Article
[15] Imre Z Ruzsa, “Sumsets and structure”, Combinatorial number theory and additive group theory (2009), p. 87-210
[16] Terence Tao, “Product set estimates for non-commutative groups”, Combinatorica 28 (2008) no. 5, p. 547-594 Article
[17] Terence Tao, “Noncommutative sets of small doubling”, European Journal of Combinatorics 34 (2013) no. 8, p. 1459-1465 Article
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