Almost primes in various settings

Abstract: Let $k \geq 3$ and let $L_i(n) = A_in + B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define ${\mathcal{P}(n) = \prod_{i=1}^k L_i(n)}$. For each $k$ it is known that $\Omega (\mathcal{P} (n) ) \leq \rho_k$ infinitely often for some integer $\rho_k$. We improve the possible values of $\rho_k$ for $4 \leq k \leq 10$ assuming $GEH$. We also show that we can take $\rho_5=14$ unconditionally. As a by-product of our approach we reprove the $\rho_3=7$ result which was previously obtained by Maynard who used techniques specifically designed for this case.