On sums of primes from Beatty sequences

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Let k ≥ 2 and α₁, β₁,..., αₖ, βₖ be reals such that the αᵢ's are irrational and greater than 1. Suppose further that some ratio αᵢ/αⱼ is irrational. We study the representations of an integer n in the form p₁ + p₂ +... + pₖ = n, where pᵢ is a prime from the Beatty sequence Bᵢ = {n ∈ N: n = [αᵢ m + βᵢ] for some m ∈ Z }.