Additive combinatorics methods in associative algebras
Confluentes Mathematici, Volume 9 (2017) no. 1, pp. 3-27.

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.

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DOI: 10.5802/cml.34
Classification: 11P70,  20D60
Keywords: Additive combinatorics, group algebras, Kneser Theorem, associative algebras, monoids
Vincent Beck 1; Cédric Lecouvey 2

1 MAPMO (UMR CNRS 7349), Université d’Orléans, F-45067 Orléans, France Fédération de Recherche Denis Poisson - CNRS 2964
2 Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083), Université François-Rabelais Tours, Parc de Grandmont, F-37200 Tours, France Fédération de Recherche Denis Poisson - CNRS 2964
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Vincent Beck; Cédric Lecouvey. Additive combinatorics methods in associative algebras. Confluentes Mathematici, Volume 9 (2017) no. 1, pp. 3-27. doi : 10.5802/cml.34. https://cml.centre-mersenne.org/articles/10.5802/cml.34/

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