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On a generalisation of Roth's theorem for arithmetic progressions and applications to sum-free subsets

Published online by Cambridge University Press:  12 June 2013

JEHANNE DOUSSE
JEHANNE DOUSSE*
Affiliation:
LIAFA, Université Paris Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France. e-mail: jehanne.dousse@liafa.univ-paris-diderot.fr
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Abstract

We prove a generalisation of Roth's theorem for arithmetic progressions to d-configurations, which are sets of the form {ni+nj+a}1 ≤ i ≤ j ≤ d with a, n1,. . .,nd ∈ $\mathbb{N}$, using Roth's original density increment strategy and Gowers uniformity norms. Then we use this generalisation to improve a result of Sudakov, Szemerédi and Vu about sum-free subsets [10] and prove that any set of n integers contains a sum-free subset of size at least logn(log(3)n)1/32772 − o(1).

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 331 - 341
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Bourgain, J.On triples in arithmetic progression. Geom. Funct. Anal. 9 (1999), pp. 968–984.CrossRefGoogle Scholar
[2]Choi, S. L. G.On a combinatorial problem in number theory. Proc. London Math. Soc. 3 (1971), pp. 629–642.CrossRefGoogle Scholar
[3]Erdős, P.Extremal problems in number theory. In Proc. Sympos. Pure Math. vol. 8 (American Mathematical Society), 1965, pp. 181–189.Google Scholar
[4]Gowers, W. T.A new proof of Szemerédi's theorem. Geom. Funct. Anal. 11 (2001), pp. 465–588.CrossRefGoogle Scholar
[5]Green, B. and Tao, T.Linear equations in primes. Ann. of Math. 171 (2010), pp. 1753–1850.CrossRefGoogle Scholar
[6]Green, B. J. Roth's theorem on progressions of length 3. Course notes.Google Scholar
[7]Heath–Brown, D. R.Integer sets containing no arithmetic progressions. J. London Math. Soc. 35 (1987), pp. 385–394.CrossRefGoogle Scholar
[8]Roth, K.On certain sets of integers. J. London Math. Soc. 28 (1953), pp. 104–109.CrossRefGoogle Scholar
[9]Ruzsa, I. Z.Sum-avoiding subsets. The Ramanujan Journal 9 (2005), pp. 77–82.CrossRefGoogle Scholar
[10]Sudakov, B., Szemerédi, E. and Vu, V.On a question of Erdös and Moser. Duke Math. J. 129 (2005), pp. 129–155.CrossRefGoogle Scholar
[11]Szemerédi, E.On sets of integers containing no $k$ elements in arithmetic progression. Acta Arith. 27 (1975), pp. 199–245.CrossRefGoogle Scholar
[12]Szemerédi, E.Integer sets containing no arithmetic progressions. Acta Math. Hungar. 56 (1990), pp. 155–158.CrossRefGoogle Scholar
[13]Tao, T. and Vu, V. H.Additive Combinatorics (Cambridge University Press, 2010).Google Scholar