Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain $L_{\infty}$-optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose $L_{\infty}$ star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.

翻译：低差异点集作为数值过程中用离散对象逼近连续对象的重要工具（例如在数值积分中）已被广泛应用。历经一个世纪的研究，点集的差异值究竟能达到多低——换言之，给定空间中点的分布能有多规则——至今仍未明确。近期利用优化与机器学习技术的研究显著改进了低差异点集的构造方法，得到了比以往所知差异值低得多的点集构型。基于这些最优构造，我们提出一种简单方法，可获得遵循（任意）输入点集相对顺序的$L_{\infty}$优化点分布。将该方法应用于维度2和3中分别最多400个和50个点的点集，我们得到的点集其$L_{\infty}$星差异比当前最优点集降低达25%，较斐波那契集等经典构造提升约50%。