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Abstract

We show that if $A$ and $B$ are subsets of the primes with positive relative lower densities $\alpha$ and $\beta$, then the lower density of $A+B$ in the natural numbers is at least $(1-o(1))\alpha/(e^{\gamma}\log \log (1/\beta))$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $\mathbb Z_m^\ast$ using techniques of Green and Green–Tao. Concerning this new problem we show that, for any square-free $m$ and any $A, B \subseteq \mathbb Z_m^\ast$ of densities $\alpha$ and $\beta$, the density of $A+B$ in $\mathbb Z_m$ is at least $(1-o(1))\alpha/(e^{\gamma} \log \log (1/\beta))$, which is asymptotically best possible when $m$ is a product of small primes. We also discuss an inverse question.