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An analogue of Vosper's theorem for extension fields

Published online by Cambridge University Press:  31 January 2017

CHRISTINE BACHOC ,
ORIOL SERRA  and
GILLES ZÉMOR
CHRISTINE BACHOC
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France. e-mail: bachoc@math.u-bordeaux.fr
ORIOL SERRA
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya and Barcelona Graduate School of Mathematics, Edifici C3, Despatx: 112, C. Jordi Girona, 1–3, 08034 Barcelona, Spain. e-mail: oriol.serra@upc.edu
GILLES ZÉMOR
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France. e-mail: zemor@math.u-bordeaux.fr
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Abstract

We are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which

\begin{linenomath}$$\dim_FST =\dim_F S+\dim_F T-1,$$\end{linenomath}
when dimFS, dimFT ⩾ 2 and dimFST ⩽ [L : F] − 2.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 423 - 452
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Bachoc, C., Serra, O. and Zémor, G. Revisiting Kneser's Theorem for Field Extensions. Combinatorica. To appear.Google Scholar
[2] Bannai, E. and Ito, T. Algebraic Combinatorics I: Association Schemes (Benjamin/Cummings Publishing Co., 1984).Google Scholar
[3] Delsarte, P. An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 197, (1973).Google Scholar
[4] Delsarte, P. Bilinear forms over a finite field, with applications to coding theory. J. Combinatorial Theory Ser. A. (3) 25 (1978), 226–241.CrossRefGoogle Scholar
[5] Delsarte, P. and Goethals, J. M. Alternating bilinear forms over GF(q). J. Combinatorial Theory Ser. A. (1) 19 (1975), 26–50.CrossRefGoogle Scholar
[6] Delsarte, P. and Levenshtein, V. Association schemes and coding theory. IEEE Trans. Inform. Theory. (6) 44 (1998), 2477–2504.CrossRefGoogle Scholar
[7] Eliahou, S. and Lecouvey, C. On linear versions of some additive theorems. Linear Multilinear Algebra. 57 (2009), 759–775.CrossRefGoogle Scholar
[8] Feng, R., Wang, Y., Ma, C. and Ma, J.. Eigenvalues of association schemes of quadratic forms. Discrete Maths. 308 (2008), 3023–3047.CrossRefGoogle Scholar
[9] Freiman, G. A. Foundations of a structural theory of set addition. Transl. Math. Monographs 37 (Amer. Math. Soc., Providence, RI, 1973).Google Scholar
[10] In memory of Yahya Ould Hamidoune, special issue of European Journal of Combinatorics, Plagne, Serra and Zémor Eds. vol. 34 (2013).Google Scholar
[11] Hamidoune, Y. O. On the connectivity of Cayley digraphs. European J. Combin. 5 (1984), 309–312.CrossRefGoogle Scholar
[12] Hamidoune, Y. O. An isoperimetric method in additive theory. J. Algebra. 179 (1996), 622–630.CrossRefGoogle Scholar
[13] Hamidoune, Y. O. Some results in additive number theory I: the critical pair theory. Acta Arith. 96 (2000), 97–119.CrossRefGoogle Scholar
[14] Hamidoune, Y. O. Some additive applications of the isoperimetric approach. Ann. Inst. Fourier. 58 (2008), fasc. 6 (2007-2036).CrossRefGoogle Scholar
[15] Hartshorne, R. Algebraic Geometry. GTM 52 (Springer).CrossRefGoogle Scholar
[16] Hou, X., Leung, K.H., and Xiang, Q. A generalization of an addition theorem of Kneser. J. Number Theory 97 (2002), 1–9.CrossRefGoogle Scholar
[17] Kemperman, J.H.B. On complexes in a semigroup. Idag. Math. 18 (1956), 247–254.Google Scholar
[18] Kneser, M. Summenmengen in lokalkompakten abelesche Gruppen. Math. Z. 66 (1956), 88–110.CrossRefGoogle Scholar
[19] Lang, S. Algebra (Springer, 3rd Edition, 2005).Google Scholar
[20] Lang, S. and Weil, A. Number of points of varieties over finite fields. Amer. J. Math. 76 (1954), 819–827.CrossRefGoogle Scholar
[21] Lecouvey, C. Plünnecke and Kneser type theorems for dimension estimates. Combinatorica (3) 34 (2014), 331–358.CrossRefGoogle Scholar
[22] MacWilliams, F.J. and Sloane, N.J.A.. The Theory of Error-Correcting Codes (North-Holland 1977).Google Scholar
[23] Mann, H. B. Addition Theorems: the Addition Theorems of Group Theory and Number Theory (Krieger Publishing Company, 1976).Google Scholar
[24] Nathanson, M.B. Additive Number Theory. Inverse problems and the geometry of sumsets. Grad. Texts in Math. 165 (Springer, 1996).CrossRefGoogle Scholar
[25] Olson, J.E. On the sum of two sets in a group. J. Number Theory 18 (1984), 110–120.CrossRefGoogle Scholar
[26] Plagne, A., Serra, O. and Zémor, G. Yahya Ould Hamidoune's mathematical journey: a critical review of his work. European J. Combin. 34 (2013), 1207–1222.CrossRefGoogle Scholar
[27] Ruzsa, I.Z. An application of graph theory to additive number theory. Sci. Ser. A Math. Sci. (N.S.) 3 (1989), 97–109.Google Scholar
[28] Schmidt, K.-U.. Symmetric bilinear forms over finite fields of even characteristic. J. Combinatorial Theory Ser. A. (8) 117 (2010), 1011–1026.CrossRefGoogle Scholar
[29] Schmidt, K.-U.. Symmetric bilinear forms over finite fields with applications to coding theory. J. Algebraic Combinatorics 42 (2015), 635–670.CrossRefGoogle Scholar
[30] Tao, T. and Vu, V. Additive Combinatorics (Cambridge University Press, 2006).CrossRefGoogle Scholar
[31] Taylor, D. E. The geometry of the classical groups Sigma Series on Pure Mathematics 9 (Heldermann, Berlin 1992).Google Scholar
[32] Vosper, G. The critical pairs of subsets of a group of prime order. J. London Math. Soc. 31 (1956), 200–205.CrossRefGoogle Scholar
[33] Wang, Y., Wang, C., Ma, C. and Ma, J.. Association Schemes of Quadratic Forms and Symmetric Bilinear Forms. J. Algebraic Combin. 17 (2003), 149–161.CrossRefGoogle Scholar