Additive problems for integers with a given number of prime divisors

The asymptotics of sums of the form $\sum_{\substack n<N \\ \omega(n)=k}\tau(|bn-a|),$ (summation over $n<N$, $\omega(n)=k$) is studied, where $\omega(n)$ is the number of distinct prime divisors of $n$, and $\tau(n)$ is the number of all divisors.