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arXiv · 0912.1842

Improving Roth's theorem in the primes

Harald Andres Helfgott·Anne de Roton
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Abstract

Let A be a subset of the primes. Let δ_P(N) = \frac{|\{n\in A: n\leq N\}|}{|\{\text{$n$ prime}: n\leq N\}|}. We prove that, if δ_P(N)\geq C \frac{\log \log \log N}{(\log \log N)^{1/3}} for N\geq N_0, where C and N_0 are absolute constants, then A\cap [1,N] contains a non-trivial three-term arithmetic progression. This improves on B. Green's result, which needs δ_P(N) \geq C' \sqrt{\frac{\log \log \log \log \log N}{\log \log \log \log N}}.

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Harald Andres Helfgott, Anne de Roton. 2009-12-09. Improving Roth's theorem in the primes. https://arxiv.org/abs/0912.1842

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