论文标题
正方形的极端特性
Extremal Property of the Square Lattice
论文作者
论文摘要
受Faulhuber-Steinerberger的2019年结果的激励，我们证明了真实的Square晶格$ \ Mathbb {Z}^2 $具有与六角形晶格$λ$相同的本地极端特性，其中晶格与自然基础域的深处差异较高的危险下增加。如果$δ$是单模型晶格空间中$ \ mathbb {z}^2 $的小扰动，请考虑$ c_r $，$ a_r $中的一组点转移到$Δ$。如果$δ$是晶格$ \ mathbb {z}^2 $相对于欧几里得公制的扰动，那么对于固定的深孔$ p $，晶格的总距离的总距离$ p $严格增加了，并且在下面的距离和其扰动之间的功能都限制在下面。此外，我们显示这种增长大约通过凸函数保存。
Motivated by a 2019 result of Faulhuber-Steinerberger, we demonstrate that the real square lattice $\mathbb{Z}^2$ exhibits the same local, extremal property as the hexagonal lattice $Λ$, where distances of lattice points from the `deep holes' of natural fundamental domains increase under perturbation. If $Δ$ is a small perturbation of $\mathbb{Z}^2$ in the space of unimodular lattices, consider $C_r$, the set of points in $A_r$ shifted to $Δ$. If $Δ$ is a perturbation of the lattice $\mathbb{Z}^2$ with respect to the Euclidean metric, then for a fixed deep hole $p$, the summed total distance of lattice points to $p$ strictly increases, and is bounded below by a function of the distance between the lattice and its perturbation. Additionally, we show this growth is approximately preserved by convex functions.
