Some ergodic theorems involving Omega function and their applications

In this paper, we build some ergodic theorems involving the function Ω , where Ω (n) denotes the number of prime factors of a natural number n counted with multiplicities. As a combinatorial application, it is shown that for any k∈ N and every A⊂ N with positive upper Banach density, there are a,d∈ N such that a,a+d,…, a+kd,a+Ω(d)∈ A.