Sums of Positive Density Subsets of the Primes
نویسنده
چکیده
We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least (1 − o(1))α/(eγ log log(1/β)) 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 Zm using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any A,B ⊆ Zm of densities α and β, the density of A+B in Zm is at least (1 − o(1))α/(eγ log log(1/β)), which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question.
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