Higher uniformity of arithmetic functions in short intervals II. Almost all intervals
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2024
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| author | Matomäki, Kaisa Radziwiłł, Maksym Shao, Xuancheng Tao, Terence Teräväinen, Joni |
| author_facet | Matomäki, Kaisa Radziwiłł, Maksym Shao, Xuancheng Tao, Terence Teräväinen, Joni |
| contents | We study higher uniformity properties of the von Mangoldt function $Λ$, the Möbius function $μ$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$.
Let $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$, $G/Γ$ a filtered nilmanifold, and $F\colon G/Γ\to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (Λ(n)-Λ^\sharp(n)) \overline{F}(g(n)Γ) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)Γ) \right| = o(H \log^{k-1} X). \]
As a consequence, we show that the short interval Gowers norms $\|Λ-Λ^\sharp\|_{U^s(X,X+H]}$ and $\|d_k-d_k^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable.
Our main new ingredients are type $II$ estimates obtained by developing a "contagion lemma" for nilsequences and then using this to "scale up" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher uniformity of arithmetic functions in short intervals II. Almost all intervals Matomäki, Kaisa Radziwiłł, Maksym Shao, Xuancheng Tao, Terence Teräväinen, Joni Number Theory 11N37, 11B30 We study higher uniformity properties of the von Mangoldt function $Λ$, the Möbius function $μ$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$, $G/Γ$ a filtered nilmanifold, and $F\colon G/Γ\to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (Λ(n)-Λ^\sharp(n)) \overline{F}(g(n)Γ) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)Γ) \right| = o(H \log^{k-1} X). \] As a consequence, we show that the short interval Gowers norms $\|Λ-Λ^\sharp\|_{U^s(X,X+H]}$ and $\|d_k-d_k^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $II$ estimates obtained by developing a "contagion lemma" for nilsequences and then using this to "scale up" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity. |
| title | Higher uniformity of arithmetic functions in short intervals II. Almost all intervals |
| topic | Number Theory 11N37, 11B30 |
| url | https://arxiv.org/abs/2411.05770 |