Plünnecke and Kneser type theorems for dimension estimates

Abstract

Given a division ring K containing the field k in its center and two finite subsets A and B of K*, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao describing the sets of small doubling in a group.

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References

  1. [1]

    S. A. Amitsur: Finite subgroups of division rings, Trans. Ams. 80 (1955), 361–386.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    G. T. Diderrich: On Kneser’s addition theorem in groups, Proc. Ams. 38 (1973), 443–451.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    S. Eliahou. and C. Lecouvey: On linear versions of some addition theorems, Linear Algebra and multilinear algebra 57 (2009), 759–775.

    Article  MATH  MathSciNet  Google Scholar 

  4. [4]

    X. D. Hou, K. H. Leung and Q Xiang: A generalization of an addition theorem of Kneser, Journal of Number Theory 97 (2002), 1–9.

    Article  MATH  MathSciNet  Google Scholar 

  5. [5]

    Y. O. Hamidoune: Some additive applications of the isoperimetric approach, Ann. Inst. Fourier 58 (2008), 2007–2036.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    Y. O. Hamidoune: Kneser theorem and some related questions, preprint (2010).

    Google Scholar 

  7. [7]

    Y. O. Hamidoune: On the connectivity of Cayley digraphs, Europ. J. Comb. 5 (1984), 309–312.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    Y. O. Hamidoune: On a subgroup contained in words with a bounded length, Discrete Math. 103 (1992), 171–176.

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

    X. D. Hou: On a vector space analogue of Kneser’s theorem, Linear Algebra and its Applications 426 (2007) 214–227.

    Article  MATH  MathSciNet  Google Scholar 

  10. [10]

    F. Kainrath: On local half-factorial orders, in: Arithmetical Properties of Commutative Rings and Monoids, Chapman & Hall/CRC, Lect. Notes. Pure Appl. Math. 241–316 (2005).

    Google Scholar 

  11. [11]

    J. H. B. Kemperman: On complexes in a semigroup, Indag. Math. 18 (1956), 247–254.

    MathSciNet  Google Scholar 

  12. [12]

    S. Lang: Algebra, Graduate Texts in Mathematics, Springer-Verlag New York Inc (2005).

    Google Scholar 

  13. [13]

    M. Madiman. A. Marcus, P. Tetali: Entropy and set cardinality inequalities for partition determined functions, with applications to sumsets, Random Structure and Algorithms (2011).

    Google Scholar 

  14. [14]

    M. B. Nathanson: Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Text in Mathematics 165, Springer-Verlag New York (1996).

    Google Scholar 

  15. [15]

    J. E. Olson: On the sum of two sets in a group, J. Number Theory 18 (1984), 110–120.

    Article  MATH  MathSciNet  Google Scholar 

  16. [16]

    G. Petridis: New proofs of Plünnecke-type estimates for product sets in groups, preprint 2011 arXiv: 1101.3507, to appear in Combinatorica.

  17. [17]

    I. Z. Ruzsa: Sumsets and structure, Combinatorial Number Theory and additive group theory, Springer New York (2009).

    Google Scholar 

  18. [18]

    T. Tao: Non commutative sets of small doublings, preprint 2011 arXiv: 11062267 (2011).

    Google Scholar 

  19. [19]

    T. Tao: Sumset and inverse sumset theorems for Shannon entropy, Combinatorics Probability and Computing 19 (2010), 603–639.

    Article  MATH  MathSciNet  Google Scholar 

  20. [20]

    T. Tao: Product set estimates for non-commutative groups, Combinatorica 28 (2009), 547–594.

    Article  Google Scholar 

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Correspondence to Cédric Lecouvey.

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Lecouvey, C. Plünnecke and Kneser type theorems for dimension estimates. Combinatorica 34, 331–358 (2014). https://doi.org/10.1007/s00493-014-2874-0

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Mathematics Subject Classification (2000)

  • 05E15
  • 12E15
  • 11P70