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Additive structures in sumsets

  • TOM SANDERS (a1)
Abstract

Suppose that A and A′ are subsets of ℤ/Nℤ. We write A+A′ for the set {a+a′:aA and a′ ∈ A′} and call it the sumset of A and A′. In this paper we address the following question. Suppose that A1,. . .,Am are subsets of ℤ/Nℤ. Does A1+· · ·+Am contain a long arithmetic progression?

The situation for m=2 is rather different from that for m ≥ 3. In the former case we provide a new proof of a result due to Green. He proved that A1+A2 contains an arithmetic progression of length roughly where α1 and α2 are the respective densities of A1 and A2. In the latter case we improve the existing estimates. For example we show that if A ⊂ ℤ/Nℤ has density then A+A+A contains an arithmetic progression of length Ncα. This compares with the previous best of Ncα2+ϵ.

Two main ingredients have gone into the paper. The first is the observation that one can apply the iterative method to these problems using some machinery of Bourgain. The second is that we can localize a result due to Chang regarding the large spectrum of L2-functions. This localization seems to be of interest in its own right and has already found one application elsewhere.

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References
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[1]Bourgain, J.On triples in arithmetic progression. Geom. Funct. Anal. 9 (1999), no. 5, 968984.
[2]Chang, M.-C.. A polynomial bound in Freiman's theorem. Duke Math. J. 113 (2002), no. 3, 399419.
[3]Croot, E.k-term arithmetic progressions in very sparse sumsets. Preprint.
[4]Green, B.J.Arithmetic progressions in sumsets. Geom. Funct. Anal. 12 (2002), no. 3, 584597.
[5]Green, B.J. Restriction and Kakeya phenomena. http://www.maths.bris.ac.uk/~mabjg.
[6]Green, B.J. and Tao, T. An inverse theorem for the Gowers U 3(G)-norm. Preprint.
[7]Rudin, W.Fourier analysis on groups. Interscience Tracts in Pure and Applied Mathematics, No. 12 (Interscience Publishers (a division of John Wiley and Sons), 1962) ix+285 pp.
[8]Ruzsa, I.Z.Generalized arithmetical progressions and sumsets. Acta Math. Hungar. 65 (1994), no. 4, 379388.
[9]Tao, T. The Roth–Bourgain theorem. http://www.math.ucla.edu/~tao.
[10]Tao, T. Lecture notes in additive combinatorics. http://www.math.ucla.edu/~tao.
[11]Tao, T. and Vu, V.Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105 (Cambridge University Press, 2006), xviii+512 pp.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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