On almost-prime $k$-tuples

Abstract: Let $\tau$ denote the divisor function and $\mathcal{H}={h_{1},...,h_{k}}$ be an admissible set. We prove that there are infinitely many $n$ for which the product $\prod_{i=1}^{k}(n+h_{i})$ is square-free and $\sum_{i=1}^{k}\tau(n+h_{i})\leq \lfloor \rho_{k}\rfloor$, where $\rho_{k}$ is asymptotic to $\frac{2126}{2853} k^{2}$. It improves a previous result of M. Ram Murty and A. Vatwani, replacing $2126/2853$ by $3/4$. The main ingredients in our proof are the higher rank Selberg sieve and Irving-Wu-Xi estimate for the divisor function in arithmetic progressions to smooth moduli.