Prime divisors in Beatty sequences

We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence ⌊ α n + β ⌋, n = 1, 2, ..., where α, β ∈ R, and α > 0 is irrational. For example, we show thatunder(∑, n ≤ N) ω (⌊ α n + β ⌋) ∼ N log log N and under(∑, n ≤ N) (- 1)^{Ω (⌊ α n + β ⌋)} = o (N), where Ω (k) and ω (k) denote the number of prime divisors of an integer k ≠ 0 counted with and without multiplicities, respectively.