论文标题
低维晶格中最短的载体
Long shortest vectors in low dimensional lattices
论文作者
论文摘要
对于coprime Integers $ n，a，b，c $，带有$ 0 <a <a <b <c <n $，我们定义$$ \ {（na \！\！\！\！\！\！\！\！\ pmod {n}，nb \！\！\！ \ leq n <n \}。 $$我们研究哪个参数$ n，a，b，c $生成点集，在集合点的依赖点之间最短距离，并将这些集合与特定形式的晶格联系起来。作为主要结果，我们提出了一个无限的晶格家族，其属性使每个晶格的最短矢量的归一化规范收敛到Hermite常数$γ_3$的平方根。我们将建筑概括至$ 4 $和$ 5 $尺寸的概括获得了类似的结果。
For coprime integers $N,a,b,c$, with $0<a<b<c<N$, we define the set $$ \{ (na \! \! \! \! \pmod{N}, nb \! \! \! \! \pmod{N}, nc \! \! \! \! \pmod{N}) : 0 \leq n < N\}. $$ We study which parameters $N,a,b,c$ generate point sets with long shortest distances between the points of the set in dependence of $N$ and relate such sets to lattices of a particular form. As a main result, we present an infinite family of such lattices with the property that the normalised norm of the shortest vector of each lattice converges to the square root of the Hermite constant $γ_3$. We obtain a similar result for the generalisation of our construction to $4$ and $5$ dimensions.
