Smooth numbers are orthogonal to nilsequences

Abstract

The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be $[y^{\prime },y]$-smooth if all of its prime factors belong to the interval $[y^{\prime },y]$. We identify suitable weights $g_{[y^{\prime },y]}(n)$ for the characteristic function of $[y^{\prime },y]$-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 $g_{[y^{\prime },y]}(n)-1$ is orthogonal to nilsequences. Our results apply in the almost optimal range ${(\log <!--FUNCTION\; APPLICATION-->N)}^K<y\le N$ of the smoothness parameter $y$, where $K\ge 2$ is sufficiently large, and to any $y^{\prime }<\min <!--FUNCTION\; APPLICATION-->(\sqrt{y},{(\log <!--FUNCTION\; APPLICATION-->N)}^c)$.

As a first application, we establish for any $y>N^{1∕\sqrt{{\log <!--FUNCTION\; APPLICATION-->}_{9}N}}$ asymptotic results on the frequency with which an arbitrary finite complexity system of shifted linear forms ${\psi }_j(n)+a_j\in \mathbb{Z}[n_1,\dots <!--FUNCTION\; APPLICATION-->,n_s]$, $1\le j\le r$, simultaneously takes $[y^{\prime },y]$-smooth values as the $n_i$ vary over integers below $N$.