On sums of primes from Beatty sequences

Let $k \ge 2$ and $α_1, β_1, ..., α_k, β_k$ be reals such that the $α_i$'s are irrational and greater than 1. Suppose further that some ratio $α_i/α_j$ is irrational. We study the representations of an integer $n$ in the form $p_1 + p_2 + ... + p_k = n,$ where $p_i$ is a prime from the Beatty sequence $\mathcal B_i = \left\{n \in \mathbb N : n = [ α_i m + β_i ] \text{for some} m \in \mathbb Z \right\}.$

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