Improving Roth's theorem in the primes

Harald Andres Helfgott, Anne de Roton

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…