论文标题
光滑的数字与尼尔序列正交
Smooth numbers are orthogonal to nilsequences
论文作者
论文摘要
本文的目的是研究没有大或小因素的整数的分布性能。将整数定义为$ [y'，y] $ - 如果其所有主要因素都属于Interval $ [y'，y] $，则将其定义为平滑。我们确定适合的权重$ g _ {[y'，y]}（n）$，用于$ [y'，y] $的特征功能 - 平滑数字，使我们能够在短暂的算术进程中为它们的分布建立强大的渐近结果。在这些等级分布属性的基础上，我们表明（$ w $ tricked版本的）$ g _ {[y'，y，y]}（n） - 1 $与nilSequences正交。我们的结果适用于平滑度参数$ y $的几乎最佳范围$（\ log n）^{k} <y \ leq n $，其中$ k \ geq 2 $足够大，而任何$ y'<\ min（\ sqrt {y} {y}，（\ log log n）^c）$。 作为第一个应用程序，我们为任何$ y> n^{1/\ sqrt {\ log_9 n}} $渐近结果逐渐限制了偏移线性的有限有限复杂性系统$ψ_j（\ mathbf {n}） \ leq r $，同时采用$ [y'，y] $ - 平滑值，因为$ n_i $的$ n_i $因$ n $以下的整数而异。
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 $ψ_j (\mathbf{n}) + a_j \in \mathbb{Z}[n_1, \dots, n_s]$, $1 \leq j \leq r$, simultaneously takes $[y',y]$-smooth values as the $n_i$ vary over integers below $N$.
