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 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 > N 1/ log 9 N asymptotic results on the frequency with which an arbitrary finite complexity system of shifted linear forms j (n) + a j [n 1 , . . ., n s ], 1 j r , simultaneously takes [y , y]-smooth values as the n i vary over integers below N.