Additive Problems with Primes from a Thin Bohr Set

Abstract

For an irrational $\alpha \in \mathbb{R}$, we consider additive problems with the set of primes satisfying $\Vert \alpha p\Vert \le \frac{1}{p^{\tau }}$ 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 $\Vert \alpha p\Vert \le \frac{1}{p^{\tau }}$ for $\tau \in (0,\frac{1}{8})$. We also consider a binary Goldbach-type problem.