论文标题
凸形域和Annuli的晶格点差异的较高矩
Higher Moments for Lattice Point Discrepancy of Convex Domains and Annuli
论文作者
论文摘要
给定一个域$ω\ subseteq \ mathbb {r}^2 $，令$ \ mathcal {d}（ω，x，x，r）$是$ \ mathbb {z}^2 $ in $ r \ ge 1 $和$ r \ ge 1 $ and $ x $ $ r \ ge $ nmin \ n mar}的晶格点的数量$$ \ MATHCAL {d}（ω，x，r）= \＃\ {（j，k）\ in \ mathbb {z}^2 ：（ j-x_1，k-x_2），k-x_2）\ inRΩ $ \ int _ {\ mathbb {t}^2} | \ mathcal {d}（ω，x，r）|^pdx $ $ p $ - 差异函数$ \ mathcal {d} $的$ p $ - Huxley在2014年表明，对于具有足够光滑边界的凸域，$ \ Mathcal {d} $的第四刻由$ \ Mathcal {O}（R^2 \ log R）界限，2019年，Colzani，Gariboldi，Gariboldi和Gigante将其扩展到更高的维度。在本文中，我们的贡献是双重的：首先，我们提供了赫x黎2014年结果的简单直接证明；其次，我们建立了$ p $ p $ thtice点的新估算值，radius $ r $的晶格点差异和任何固定的厚度$ 0 <t <1 $ for $ p \ ge 2。
Given a domain $Ω\subseteq \mathbb{R}^2$, let $\mathcal{D}(Ω,x,R)$ be the number of lattice points from $\mathbb{Z}^2$ in $RΩ-x$, for $R \ge 1$ and $x\in \mathbb{T}^2$, minus the area of $RΩ$: $$\mathcal{D}(Ω,x,R) = \# \{ (j,k) \in \mathbb{Z}^2 :(j-x_1,k-x_2) \in RΩ\} - R^2|Ω|.$$ We call $\int_{\mathbb{T}^2}|\mathcal{D}(Ω,x,R)|^pdx$ 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 $p\ge 2.$
