Local divisor correlations in almost all short intervals

Let $k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x<n\leq x+H_1}d_k(n)d_l(n+h)$. More precisely, let $\varepsilon>0$ be a small fixed number and let $\Phi(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{\Phi(x)}$ and $(\log x)^{1000k\log k}\leq H_2\leq H_1^{1-\varepsilon }$, the expected asymptotic formula holds for almost all $x\in[X,2X]$ and almost all $1\leq h\leq H_2$.