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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.16892 |
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| _version_ | 1866908525691666432 |
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| author | Matthiesen, Lilian Wang, Mengdi |
| author_facet | Matthiesen, Lilian Wang, Mengdi |
| contents | 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 $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16892 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smooth numbers are orthogonal to nilsequences Matthiesen, Lilian Wang, Mengdi Number Theory 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 $N$. |
| title | Smooth numbers are orthogonal to nilsequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2211.16892 |