Additive combinatorics methods in associative algebras
Confluentes Mathematici, Tome 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.

Reçu le : 2015-06-23
Révisé le : 2016-09-20
Accepté le : 2017-02-19
Publié le : 2017-09-14
DOI : https://doi.org/10.5802/cml.34
Classification : 11P70,  20D60
Mots clés: Additive combinatorics, group algebras, Kneser Theorem, associative algebras, monoids
@article{CML_2017__9_1_3_0,
     author = {Vincent Beck and C\'edric Lecouvey},
     title = {Additive combinatorics methods in associative algebras},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {9},
     number = {1},
     year = {2017},
     pages = {3-27},
     doi = {10.5802/cml.34},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2017__9_1_3_0/}
}
Beck, Vincent; Lecouvey, Cédric. Additive combinatorics methods in associative algebras. Confluentes Mathematici, Tome 9 (2017) no. 1, pp. 3-27. doi : 10.5802/cml.34. https://cml.centre-mersenne.org/item/CML_2017__9_1_3_0/

[1] Christine Bachoc; Oriol Serra; Gilles Zémor Revisiting Kneser’s Theorem for Field Extensions (2015) (https://arxiv.org/abs/1510.01354)

[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, Tome 38 (1973) no. 3, pp. 443-451

[4] Shalom Eliahou; Cédric Lecouvey On linear versions of some addition theorems, Linear and multilinear algebra, Tome 57 (2009) no. 8, pp. 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, Tome 5 (1984) no. 4, pp. 309-312 | Article

[8] Xiang-dong Hou On a vector space analogue of Kneser’s theorem, Linear Algebra and its Applications, Tome 426 (2007) no. 1, pp. 214-227 | Article

[9] Xiang-Dong Hou; Ka Hin Leung; Qing Xiang A generalization of an addition theorem of Kneser, Journal of Number Theory, Tome 97 (2002) no. 1, pp. 1-9 | Article

[10] Florian Kainrath On local half-factorial orders, Arithmetical properties of commutative rings and monoids, Tome 241 (2005), pp. 316-324 | Article

[11] Cédric Lecouvey Plünnecke and Kneser type theorems for dimension estimates, Combinatorica, Tome 34 (2014) no. 3, pp. 331-358 | Article

[12] Diego Mirandola; Gilles Zémor Critical pairs for the product singleton bound, IEEE Transactions on Information Theory, Tome 61 (2015) no. 9, pp. 4928-4937 | Article

[13] Melvyn B Nathanson Additive Number Theory: Inverse Problems and the Geometry of Sumsets Tome 165, Springer, 1996, p. ALL-ALL

[14] John E Olson On the sum of two sets in a group, Journal of Number Theory, Tome 18 (1984) no. 1, pp. 110-120 | Article

[15] Imre Z Ruzsa Sumsets and structure, Combinatorial number theory and additive group theory (2009), pp. 87-210

[16] Terence Tao Product set estimates for non-commutative groups, Combinatorica, Tome 28 (2008) no. 5, pp. 547-594 | Article

[17] Terence Tao Noncommutative sets of small doubling, European Journal of Combinatorics, Tome 34 (2013) no. 8, pp. 1459-1465 | Article