Additive Problems with Primes from a Thin Bohr Set

For an irrational $\alpha\in \mathbb{R}$, we consider additive problems with the set of primes satisfying $\lVert\alpha p\rVert\leq \frac{1}{p^\tau}$ for some fixed $\tau>0$. In particular, we show that there exist infinitely many non-trivial three-term arithmetic progressions in the set of primes satisfying $\lVert \alpha p\rVert\leq \frac{1}{p^\tau}$ for $\tau\in(0, \tfrac18)$. We also consider a binary Goldbach-type problem.