Additive Problems with Primes from a Thin Bohr Set

TL;DR

This paper investigates additive problems involving primes constrained by a Diophantine approximation condition with an irrational number, establishing the existence of infinitely many 3-term arithmetic progressions within this set.

Contribution

It proves the existence of infinitely many 3-term arithmetic progressions in primes satisfying a specific irrational approximation condition, for the first time in this context.

Findings

Existence of infinitely many 3-term arithmetic progressions in the prime set.

Results hold for a range of au in (0, 1/8).

Addresses a binary Goldbach-type problem.

Abstract

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.