Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its generating vector $\boldsymbol{z} \in \mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of "good" rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. In the paper [5], we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in [5].

翻译：Lattice 规则是研究得最为突出的准Monte Carlo 近似多变构件的方法之一。 生成矢量 $\boldsymbol{z} $\ mathbb ⁇ s $\ mathbb ⁇ s 和点数 ~ N$ 来充分指定了约合美元元元元元元元元元元元元元元元元元元元值的一至一元值规则。 虽然在存在“ 好”一等-1美元规则方面有许多结果, 但对于维量值为$s\ Ge 3 的生成良好矢量没有明确的构建。 这就是为什么通常使用计算机搜索算法的原因。 在文件 [5]中,我们展示了一种逐个组件数字逐位数(CBCC-DBDDD)的构造, 用于生成一等量值函数的优质矢量, 整合在加权的 Korobov 类中的功能。 但是, 该文件的结果仅限于产品重量。 在本文中, 我们将这一结果概括为任意正积重量,, 从而回答 [ 5] 中提出的一个开放的问题。