论文标题
改善了Möbius函数的二次高级统一性
Improved quadratic Gowers uniformity for the Möbius function
论文作者
论文摘要
我们证明，$ \ | | | _ {u^3（[n]）} \ ll_ {a}^{\ text {\ text {ineff}} \ log^{ - a} { - a}（n）$ $ $ $ $ $ \ | | |λ-λ_q\ | __q \ | _________ {u^u^3（n] n（n）} \ ll_ {a}^{\ text {ineff}} \ log^{ - a}（n）$$对于任何$ a> 0 $，其中$λ_q$是von mangoldt函数的近似值，并将在下面定义，并在tao-teräväinen（20221）的范围内改进。结果，除其他外，我们还有以下内容：$$ \ mathbb {e} _ {x，y \ in [n]，x + 3y \ in [n]}λ（x）λ（x + y）λ（x + y）λ（x + 2y）λ（x + 2y）λ（ + 3y）λ（x + 3y）= \ m nerive = \ nher $ \ mathfrak {s} $是配置$（x，x + y，x + 2y，x + 3y）$的单数系列。实际上，我们表明$ |μ-μ_{siegel} \ | _ {u^3（[n]）} \ ll \ exp（-o（-o log^{1/c}（n）） \ exp（-o（\ log^{1/c}（n）））$$其中$μ_{siegel} $和$λ_{siegel} $分别是$μ$的近似值，和$λ$，代表$ $ $ $ $ $ $和上述文章中的siegel零贡献。为此，我们使用$ U^3 $逆定理的改进，我们遵循Green and Tao（2007）的方法，选择使用``老式的''方法在两步尼尔曼群岛上进行等距的nilmanifold，这也是Green and Tao（2017年），以及Anevieration and timier timier timier（2010年）。 $ \ mathbb {z}/n \ mathbb {z} $在应用程序中实现了准polynomial类型范围。
We demonstrate that $$\|μ\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ $$\|Λ- Λ_Q\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ for any $A > 0$ where $Λ_Q$ is an approximant to the von Mangoldt function and will be defined below, improving upon a bound of Tao-Teräväinen (2021). As a consequence, among other things, we have the following: $$\mathbb{E}_{x, y \in [N], x + 3y \in [N]} Λ(x)Λ(x + y)Λ(x + 2y)Λ(x + 3y) = \mathfrak{S} + O_A(\log^{-A}(N))$$ where $\mathfrak{S}$ is the singular series for the configuration $(x, x + y, x + 2y, x + 3y)$. In fact, we show that $$\|μ- μ_{Siegel}\|_{U^3([N])} \ll \exp(-O(\log^{1/C}(N)))$$ $$\|Λ- Λ_{Siegel}\|_{U^3([N])} \ll \exp(-O(\log^{1/C}(N)))$$ where $μ_{Siegel}$ and $Λ_{Siegel}$ are approximants of $μ$, and $Λ$, respectively, representing the Siegel zero contribution of $μ$ and are defined in the above article. To do so, we use an improvement of the $U^3$ inverse theorem due to Sanders and we follow the approach of Green and Tao (2007), opting to use the ``old-fashioned" approach to equidistribution on two-step nilmanifolds which was also considered by Green and Tao (2017), and by Gowers and Wolf (2010). To the author's knowledge, this is the first time that quadratic Fourier analysis over $\mathbb{Z}/N\mathbb{Z}$ has achieved quasi-polynomial type bounds in applications.
