A short-interval Hildebrand-Tenenbaum theorem

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $\nu$ distinct prime divisors: $\pi_{\nu}(x) \sim x \delta_{\nu}(x)$. Here we consider the restricted count $\pi_{\nu}(x,y)$ for integers lying in the short interval $(x,x+y]$. In this setting, we show that for any $\varepsilon >0$, the asymptotic equivalence $$\pi_{\nu}(x,y) \sim y \delta_{\nu}(x)$$ holds uniformly over all $1 \le \nu \le (\log x)^{1/3}/(\log \log x)^2$ and all $x^{17/30 + \varepsilon} \leq y \leq x$. The methods also furnish mean upper bounds for the $k$-fold divisor function $\tau_k$ in short intervals, with strong uniformity in $k$.