For an irrational α R, we consider additive problems with the set of primes satisfying αp 1p^τ for some fixed τ>0. In particular, we show that there exist infinitely many non-trivial three-term arithmetic progressions in the set of primes satisfying αp 1p^τ for τ (0, 18). We also consider a binary Goldbach-type problem.
Saurabh Jain (Sat,) studied this question.