Year: 2025
Type: article
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 ′ , 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 ψ j (n) + a j ∈ [ޚn 1 , . . ., n s ], 1 ⩽ j ⩽ r , simultaneously takes [y ′ , y]-smooth values as the n i vary over integers below N.
Source: Algebra & Number Theory
Authors: Lilian Matthiesen, Mengdi Wang
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4414012967