Additive Problems with Primes from a Thin Bohr Set

For an irrational α∈ R, we consider additive problems with the set of primes satisfying αp≤ 1/p^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≤ 1/p^τ for τ∈(0, 18). We also consider a binary Goldbach-type problem.