Improving Roth's theorem in the primes

Harald Andrès Helfgott (1) , Anne de Roton (2)

1 ENS-PSL - École normale supérieure - Paris
2 IECN - Institut Élie Cartan de Nancy

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}.

Domaines: Théorie des nombres [math.NT]

Harald Andrès Helfgott, Anne de Roton. Improving Roth's theorem in the primes. International Mathematics Research Notices, 2011, 4, pp.767-783. ⟨10.1093/imrn/rnq108⟩. ⟨hal-01280111⟩