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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: [email protected]
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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).

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

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