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.
Revised:
Accepted:
Published online:
Keywords: Additive combinatorics, group algebras, Kneser Theorem, associative algebras, monoids
Vincent Beck 1; Cédric Lecouvey 2
CC-BY-NC-ND 4.0
@article{CML_2017__9_1_3_0,
author = {Vincent Beck and C\'edric Lecouvey},
title = {Additive combinatorics methods in associative algebras},
journal = {Confluentes Mathematici},
pages = {3--27},
publisher = {Institut Camille Jordan},
volume = {9},
number = {1},
year = {2017},
doi = {10.5802/cml.34},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.34/}
}
TY - JOUR AU - Vincent Beck AU - Cédric Lecouvey TI - Additive combinatorics methods in associative algebras JO - Confluentes Mathematici PY - 2017 SP - 3 EP - 27 VL - 9 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.34/ DO - 10.5802/cml.34 LA - en ID - CML_2017__9_1_3_0 ER -
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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