AI 中文总结
本研究从p次分圆域子域的代数整数环中特定ℤ-模族出发构造代数格，计算其中心密度下界，得到2、3、5维最优填充密度的代数格，助力经典球体填充问题研究。
AI 中文摘要
经典球体填充问题至今尚未解决，该问题旨在确定大量相同球体的最密堆积方式。在部分球体填充中，球心构成欧几里得格，即ℝⁿ中的离散加法子群。代数数域整数环中的自由ℤ-模通过典范嵌入可生成代数格。本研究从p次分圆域子域的代数整数环中特定ℤ-模族出发，提出代数格的新构造方法，其中p为素数。在该框架下，我们计算了这些代数格的中心密度下界，并构造出在2、3、5维具有已知最优填充密度的代数格。
英文摘要
The classical sphere packing problem, which remains unsolved, consists of determining how densely a large number of identical spheres can be packed together. In some sphere packings, the centers of the spheres in a sphere packing form a Euclidean lattice, which is a discrete additive subgroup of $\mathbb{R}^n$. Free $\mathbb{Z}$-modules in the ring of integers of an algebraic number field yield algebraic lattices via the canonical embedding. In this work, we present new constructions of algebraic lattices from certain families of $\mathbb{Z}$-modules in the ring of algebraic integers of subfields of the $p$-th cyclotomic field, where $p$ is a prime number. Within this framework, we compute lower bounds for the center density of these algebraic lattices and construct algebraic lattices having the best known packing density in dimensions 2, 3, and 5.
