On sums of primes from Beatty sequences

Let $k\ge 2$ and ${\alpha }_1,{\beta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_k,{\beta }_k$ be reals such that the ${\alpha }_i$’s are irrational and greater than 1. Suppose further that some ratio ${\alpha }_i\mathrm{/}{\alpha }_j$ is irrational. We study the representations of an integer $n$ in the form $$p_1+p_2+\mathrm{.}\mathrm{.}\mathrm{.}+p_k=n,$$ where $p_i$ is a prime from the Beatty sequence $${\mathcal{B}}_i=\{n\in \mathbb{N}:n=[{\alpha }_{i}m+{\beta }_i]\text{for some}m\in \mathbb{Z}\}\mathrm{.}$$