Improving Roth's theorem in the primes

Abstract : Let A be a subset of the primes. Let δP (N) = |{n ∈ A : n ≤ N }|/ |{n prime : n ≤ N }| be the relative density of A in the primes. We prove that if δP(N) ≥ C(log log log N)/(log log N)^{1/3} for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N] contains a non-trivial three-term arithmetic progression. This improves on Green's result [4], which needs δP(N) ≥ C'(log log log log log N/log log log log N)^{1/2}.
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Harald Helfgott, Anne de Roton. Improving Roth's theorem in the primes. International Mathematics Research Notices, Oxford University Press (OUP), 2011, 4, pp.767-783. ⟨10.1093/imrn/rnq108⟩. ⟨hal-01280111⟩

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