A note on additive properties of dense subsets of sifted sequences

In this paper we show that if A is a subset of the primes with positive lower relative density δ then A+A must have positive lower density at least C1δ/ log log(1/δ) in the natural numbers. Our argument uses the techniques developed by the author and I.Ruzsa in their work on additive properties of dense subsequences of sufficiently sifted sequences. The result is optimal and improves on recent work of K.Chipeniuk & M.Hamel. We continue by proving several similar results, by successively replacing the sequence of primes by the sequence of sums of two squares, by the sequence of those integers n that are such that n and n+ 1 are both a sum of two squares and finally by the sequence of primes p that are such that p+ 1 is a sum of two squares. The second part of this paper contains a heuristical argument that leads to several conjectures concerning the existence of k-term arithmetic progressions within these sequences. We conclude with some conjectures belonging to the Ramsey part of additive number theory.