🤖 AI Summary
This work addresses the construction of scalable point sets in the high-dimensional unit cube whose two-dimensional coordinate projections all exhibit quasi-uniform distribution. For this purpose, it pioneers a synthesis of algebraic number theory and quasi-Monte Carlo methods, leveraging rational directions on the projective line to unify the parametrization of two novel constructions: Kronecker sequences over cubic number fields and nested rank-1 lattice rules combining real quadratic fields with $p$-adic embeddings. Through techniques from algebraic norm estimation, dual Diophantine approximation, and $p$-adic analysis, the resulting point sets achieve optimal-order lower bounds on separation radii and upper bounds on covering radii for all two-dimensional projections, uniformly across any number of points. Moreover, their mesh ratios remain uniformly bounded, thereby ensuring globally consistent geometric quality and excellent scalability.
📝 Abstract
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.