Smooth numbers are orthogonal to nilsequences

Matthiesen, Lilian; Wang, Mengdi

doi:10.2140/ant.2025.19.1881

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Smooth numbers are orthogonal to nilsequences

Lilian Matthiesen and Mengdi Wang

Vol. 19 (2025), No. 10, 1881–1946

DOI: 10.2140/ant.2025.19.1881

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 ${(\log N)}^{K} < y \leq N$ of the smoothness parameter $y$ , where $K \geq 2$ is sufficiently large, and to any $y^{'} < \min (\sqrt{y}, {(\log N)}^{c})$ .

As a first application, we establish for any $y > N^{1 ∕ \sqrt{\log_{9} N}}$ asymptotic results on the frequency with which an arbitrary finite complexity system of shifted linear forms $ψ_{j} (n) + a_{j} \in ℤ [n_{1}, \dots, n_{s}]$ , $1 \leq j \leq r$ , simultaneously takes $[y^{'}, y]$ -smooth values as the $n_{i}$ vary over integers below $N$ .

Keywords

smooth numbers, multiplicative functions, nilsequences, Gowers uniformity norms, higher-order Fourier analysis

Mathematical Subject Classification

Primary: 11N37

Secondary: 11B30, 11D04, 11L15, 11N25

Milestones

Received: 10 December 2022

Revised: 17 October 2024

Accepted: 5 December 2024

Published: 5 September 2025

Authors

Lilian Matthiesen
Mathematisches Institut Georg-August-Universität Göttingen Göttingen Germany

Mengdi Wang
Institute of Mathematics École Polytechnique Fédérale de Lausanne Lausanne Switzerland

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Vol. 19, No. 10, 2025