具有拟均匀二维投影的点序列的代数构造
Algebraic constructions of point sequences with quasi-uniform two-dimensional projections
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中文总结 AI 辅助
该研究针对计算机实验空间填充设计的需求，利用数域等代数工具构造了两类二维投影拟均匀的可扩展点集，证明其投影网格比一致有界，还提出了相关极小极大问题，为空间填充点集提供了代数构造方法。
中文摘要 AI 辅助
受计算机实验的序贯空间填充设计的启发，我们研究d维单位立方体中可扩展点集的代数构造，这类点集的所有二维坐标投影均为拟均匀的。我们的两种构造在参数化上具有共性，均基于射影直线$\boldsymbol{\text{P}}^1(\boldsymbol{\text{Q}})$上不同有理方向的有限构型。第一种构造利用三次数域，得到显式克罗内克序列，其所有二维坐标投影的网格比在长度$N\boldsymbol{\text{≥}}2$的每个初始段上均保持一致有界。第二种构造利用实二次数域、分裂素数及兼容的p进嵌入，得到嵌套的秩1格设计，该设计在每个嵌套层级均具有相同的均匀投影性质。证明过程结合了代数范数估计与联立及对偶丢番图逼近之间的转移原理，得到分离半径的下界与覆盖半径的上界，二者在点的数量上均达到最优阶，且在所有坐标对上一致。我们还研究了有理射影系数的选择如何影响所得的网格比，这引出了$\boldsymbol{\text{P}}^1(\boldsymbol{\text{Q}})$上有限构型的极小极大问题，即寻求最小化所有二维坐标投影的最大网格比。这些构造提供了具有均匀控制的二元几何的可扩展点集。
英文摘要
Motivated by sequential space-filling designs for computer experiments, we study algebraic constructions of extensible point sets in the $d$-dimensional unit cube whose two-dimensional coordinate projections are all quasi-uniform. Our two constructions share a common parametrization in terms of finite configurations of distinct rational directions on the projective line $\mathbb{P}^1(\mathbb{Q})$. First, using a cubic number field, we construct explicit Kronecker sequences for which the mesh ratios of all two-dimensional coordinate projections remain uniformly bounded over every initial segment of length $N\ge 2$. Second, using a real quadratic field, a split prime, and a compatible $p$-adic embedding, we construct nested rank-1 lattice designs with the same uniform projection property at every nesting level. The proofs combine algebraic norm estimates with transference principles between simultaneous and dual Diophantine approximation, yielding lower bounds for the separation radii and upper bounds for the covering radii, both of optimal order in the number of points, uniformly over all coordinate pairs. We also investigate how the choice of rational projective coefficients affects the resulting mesh ratios. This leads to a minimax problem for finite configurations on $\mathbb{P}^1(\mathbb{Q})$, in which one seeks to minimize the maximum mesh ratio over all two-dimensional coordinate projections. These constructions provide extensible point sets with uniformly controlled bivariate geometry.
