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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
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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