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 gy′,y 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 gy′,y − 1 is orthogonal to nilsequences. Our results apply in the almost optimal range (log N)K < y ⩽ N of the smoothness parameter y, where K ⩾ 2 is sufficiently large, and to any y′ < min(√y, (log N)c ). As a first application, we establish for any y > N1/√log9N asymptotic results on the frequency with which an arbitrary finite complexity system of shifted linear forms ψj (n)+aj ∈ ℤ[n1, …, ns ], 1 ⩽ j ⩽ r, simultaneously takes [y′, y]-smooth values as the ni vary over integers below N.