Smooth numbers are orthogonal to nilsequences

The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be $[y',y]$-smooth if all of its prime factors belong to the interval $[y',y]$. We identify suitable weights $g_{[y',y]}(n)$ for the characteristic function of $[y',y]$-smooth numbers that allow us to establish strong asymptotic results on their distribution in short arithmetic progressions. Building on these equidistribution properties, we show that (a $W$-tricked version of) the function $g_{[y',y]}(n) - 1$ is orthogonal to nilsequences. Our results apply in the almost optimal range $(\log N)^{K} < y \leq N$ of the smoothness parameter $y$, where $K \geq 2$ is sufficiently large, and to any $y' < \min(\sqrt{y}, (\log N)^c)$. As a first application, we establish for any $y> N^{1/\sqrt{\log_9 N}}$ asymptotic results on the frequency with which an arbitrary finite complexity system of shifted linear forms $\psi_j (\mathbf{n}) + a_j \in \mathbb{Z}[n_1, \dots, n_s]$, $1 \leq j \leq r$, simultaneously takes $[y',y]$-smooth values as the $n_i$ vary over integers below $N$.