×

Additive combinatorics methods in associative algebras. (English) Zbl 1426.11120

Summary: 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.

MSC:

11P70 Inverse problems of additive number theory, including sumsets
16S34 Group rings
20D60 Arithmetic and combinatorial problems involving abstract finite groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Christine Bachoc, Oriol Serra & Gilles Zémor, “Revisiting Kneser’s Theorem for Field Extensions”, , 2015 · Zbl 1405.11134
[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 Society38 (1973) no. 3, p. 443-451 · Zbl 0266.20041
[4] Shalom Eliahou & Cédric Lecouvey, “On linear versions of some addition theorems”, Linear and multilinear algebra57 (2009) no. 8, p. 759-775 · Zbl 1263.11015 · doi:10.1080/03081080802018083
[5] William Fulton, Algebraic curves, The Benjamin/Cummings Publishing Company, Inc., 1969 · Zbl 0181.23901
[6] David J Grynkiewicz, “Structural Additive Theory, Dev. Math” 2013 · Zbl 1368.11109
[7] Yahya Ould Hamidoune, “On the connectivity of Cayley digraphs”, European Journal of Combinatorics5 (1984) no. 4, p. 309-312 · Zbl 0561.05028 · doi:10.1016/S0195-6698(84)80034-7
[8] Xiang-dong Hou, “On a vector space analogue of Kneser’s theorem”, Linear Algebra and its Applications426 (2007) no. 1, p. 214-227 · Zbl 1132.12003 · doi:10.1016/j.laa.2007.04.019
[9] Xiang-Dong Hou, Ka Hin Leung & Qing Xiang, “A generalization of an addition theorem of Kneser”, Journal of Number Theory97 (2002) no. 1, p. 1-9 · Zbl 1034.11020 · doi:10.1006/jnth.2002.2793
[10] Florian Kainrath, “On local half-factorial orders”, Arithmetical properties of commutative rings and monoids241 (2005), p. 316-324 · Zbl 1069.11047 · doi:10.1201/9781420028249.ch21
[11] Cédric Lecouvey, “Plünnecke and Kneser type theorems for dimension estimates”, Combinatorica34 (2014) no. 3, p. 331-358 · Zbl 1324.05196 · doi:10.1007/s00493-014-2874-0
[12] Diego Mirandola & Gilles Zémor, “Critical pairs for the product singleton bound”, IEEE Transactions on Information Theory61 (2015) no. 9, p. 4928-4937 · Zbl 1359.94724 · doi:10.1109/TIT.2015.2450207
[13] Melvyn B Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets 165, Springer, 1996 · Zbl 0859.11003
[14] John E Olson, “On the sum of two sets in a group”, Journal of Number Theory18 (1984) no. 1, p. 110-120 · Zbl 0524.10043 · doi:10.1016/0022-314X(84)90047-7
[15] Imre Z Ruzsa, “Sumsets and structure”, Combinatorial number theory and additive group theory (2009), p. 87-210 · Zbl 1221.11026
[16] Terence Tao, “Product set estimates for non-commutative groups”, Combinatorica28 (2008) no. 5, p. 547-594 · Zbl 1254.11017 · doi:10.1007/s00493-008-2271-7
[17] Terence Tao, “Noncommutative sets of small doubling”, European Journal of Combinatorics34 (2013) no. 8, p. 1459-1465 Published by the CNRS - UMR 5208 and the CNRS - UMR 5669 of eISSN : 1793-7434 · Zbl 1371.11032 · doi:10.1016/j.ejc.2013.05.028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.