  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. Removing small prime factors $p<y'$ and studying $[y',y]$-smooth
numbers instead of $y$-smooth numbers is integral to obtaining equidistribution
in short arithmetic progressions when $y$ is small. 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 for $y'<y^2$ of the shape $y' =
(\log N)^{K'}$.
  As an application, we establish for any $y> N^{1/\sqrt{\log_9 N}}$ asymptotic
results on the number of solutions in $[y',y]$-smooth numbers to arbitrary
finite complexity systems of linear equations in the integers below $N$.

Number Theory