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Freiman–Ruzsa-type theory for small doubling constant

Published online by Cambridge University Press:  01 March 2009

HANSHENG DIAO
HANSHENG DIAO*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail: hansheng@mit.edu
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Abstract

In this paper, we study the linear structure of sets A ⊂ with doubling constant σ(A) < 2, where σ(A):=|A+A|/|A|. In particular, we show that A is contained in a small affine subspace. We also show that A can be covered by at most four shifts of some subspace V with |V| ≤ |A|. Finally, we classify all binary sets with small doubling constant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 269 - 276
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Cohen, G. and Zémor, G.Subsets sums and coding theory. Astérisque 258 (1999), 327–339.Google Scholar
[2]Deshouillers, J. M., Hennecart, F. and Plagne, A. On small sumsets in ()n. Combinatorica 24 (1) (2004), 53–68.CrossRefGoogle Scholar
[3]Freiman, G.Foundations of a structural theory of set addition. Trans. Math. Monogr. 37 (American Mathematical Society, 1973).Google Scholar
[4]Freiman, G.Structure theory of set addition. Astérisque 258 (1999), 1–33.Google Scholar
[5]Green, B. and Tao, T. Freiman's theorem in finite fields via extremal set theory. Available at http://www.arxiv.org/abs/math/0703668Google Scholar
[6]Green, B. and Ruzsa, I. Z.Freiman's theorem in an arbitrary abelian group. J. London Math. Soc. 75 (2007), 163–175.CrossRefGoogle Scholar
[7]Green, B. and Tao, T. A note on the Freiman and Balog–Szemerédi–Gowers theorems in finite fields. Available at http://www.arxiv.org/abs/math.CO/0701585.Google Scholar
[8]Kemperman, J. H. B.On small sumsets in an abelian group. Acta Mathematica 103 (1960), 63–88.CrossRefGoogle Scholar
[9]Lev, V.Critical pairs in abelian groups and Kemperman's structure theorem. Int. J. Number Theory 2 (2006), 379–396.CrossRefGoogle Scholar
[10]Tao, T. and Vu, V.Additive Combinatorics (Cambridge University Press, 2006) 2006.CrossRefGoogle Scholar
[11]Zémor, G.Subset sums in binary spaces. European J. Combin. 13 (1992), 221–230.CrossRefGoogle Scholar