论文标题
原始点包装
Primitive point packing
论文作者
论文摘要
当$ d $ d $ d $二维整数晶格$ \ mathbb {z}^d $相对出色时，这是原始的。当两个原始点相反时，两个原始点是彼此的倍数，因此，我们考虑了晶格中原始点的一半，那些第一个非零坐标为正的点。我们解决了包装问题，该问题要求最多可能的任何给定坐标总和的绝对值到固定整数$ k $。在几何学，数理论和组合学的交集中，我们给出了这种结果的几个后果。特别是，我们获得了HyperCube $ [0，k]^d $中包含的晶格界限最大直径的显式表达式，并且对该超管中的任何晶状体多层构想的指出。
A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the primitive points within the lattice, the ones whose first non-zero coordinate is positive. We solve the packing problem that asks for the largest possible number of such points whose absolute values of any given coordinate sum to at most a fixed integer $k$. We present several consequences of this result at the intersection of geometry, number theory, and combinatorics. In particular, we obtain an explicit expression for the largest possible diameter of a lattice zonotope contained in the hypercube $[0,k]^d$ and, conjecturally of any lattice polytope in that hypercube.
