Text

Given a domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$, let $\mathcal{D}(\mathrm{\Omega },x,R)$ be the number of lattice points from ${\mathbb{Z}}^2$ in $R\mathrm{\Omega }-x$, for $R\ge 1$ and $x:=(x_1,x_2)\in {\mathbb{T}}^2$, minus the area of RΩ:$\mathcal{D}(\mathrm{\Omega },x,R)=\mathrm{\#}\{(j,k)\in {\mathbb{Z}}^2:(j-x_1,k-x_2)\in R\mathrm{\Omega }\}-R^2|\mathrm{\Omega }|.$ We call ${\int }_{{\mathbb{T}}^2}|\mathcal{D}(\mathrm{\Omega },x,R){|}^{p}dx$ the p-th moment of the discrepancy function $\mathcal{D}$. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of $\mathcal{D}$ is bounded by $\mathcal{O}(R^2\log R)$, and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions.

In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p-th moments of lattice point discrepancy of annuli of radius R, and any fixed thickness $0<t<1$ for $p\ge 2$.

Video

For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q.

中文翻译：

---

凸域和环的晶格点差异的更高矩

文本

给定一个域$\mathrm{\Omega }\subseteq {\mathbb{R}}^2$， 让$\mathcal{D}(\mathrm{\Omega },X,R)$是格点的数量${\mathbb{Z}}^2$在$R\mathrm{\Omega }-X$， 为了$R\ge 1$和$X：=(X_1,X_2)\in {\mathbb{吨}}^2$, 减去R Ω的面积：$\mathcal{D}(\mathrm{\Omega },X,R)=\mathrm{\#}\{(j,ķ)\in {\mathbb{Z}}^2：(j-X_1,ķ-X_2)\in R\mathrm{\Omega }\}-R^2|\mathrm{\Omega }|.$我们称之为${\int }_{{\mathbb{吨}}^2}|\mathcal{D}(\mathrm{\Omega },X,R){|}^{p}dX$差异函数的p时刻$\mathcal{D}$. 2014 年，Huxley 表明，对于边界足够光滑的凸域，$\mathcal{D}$由$\mathcal{○}(R^2\mathrm{日志}R)$，并且在 2019 年，Colzani、Gariboldi 和 Gigante 将这一结果扩展到更高的维度。

在本文中，我们的贡献是双重的：首先，我们提供了赫胥黎 2014 年结果的简单直接证明；其次，我们对半径为R的环的晶格点差异的p阶矩和任何固定厚度建立新的估计$0<吨<1$为了$p\ge 2$.

视频

有关本文的视频摘要，请访问 https://youtu.be/YWIe1IBIi9Q。