2022/11/30 by Matthiesen, Lilian, Wang, Mengdi · 2 citations
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2211.16892
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> N1/√(log9 N) 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.