Additive Problems with Primes from a Thin Bohr Set

Abstract

Abstract For an irrational alpha element of double struck upper R α ∈ R $\alpha\in {{\mathbb{R}}}$ and any beta element of double struck upper R β ∈ R $\beta\in {{\mathbb{R}}}$ , we consider additive problems with the set of primes satisfying StartMetric alpha p plus beta EndMetric less than or equals 1 divided by p Superscript tau ‖ α p + β ‖ ≤ 1 / p τ $\left\lVert\alpha p+\beta\right\rVert\leq {1}/{p^\tau}$ for some fixed tau greater than 0 τ > 0 $\tau\gt0$ . In particular, we show that there exist infinitely many non-trivial three-term arithmetic progressions in the set of primes satisfying StartMetric alpha p plus beta EndMetric less than or equals 1 divided by p Superscript tau ‖ α p + β ‖ ≤ 1 / p τ $\left\lVert\alpha p+\beta\right\rVert\leq {1}/{p^\tau}$ for tau element of left parenthesis 0 comma 1 divided by 8 right parenthesis τ ∈ ( 0 , 1 / 8 ) $\tau\in(0, 1/8)$ . We also consider a binary Goldbach-type problem.