This paper is a follow-up to the recent paper "A note on isotropic discrepancy and spectral test of lattice point sets" [J. Complexity, 58:101441, 2020]. We show that the isotropic discrepancy of a lattice point set is at most $d \, 2^{2(d+1)}$ times its spectral test, thereby correcting the dependence on the dimension $d$ and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test -- and with it the isotropic discrepancy -- is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.

翻译：本文是最近一篇论文“关于异位差异和拉蒂点各组光谱测试的说明”[J. Complicity, 58:101441, 2020]。我们显示,一个拉蒂点设定的异位差异是其光谱测试的两倍之多,从而纠正了对维值的依赖性,也纠正了上述文件Theorem 2 的上界证据的不准确性。主要任务是将单元立方体中包含的圆锥体的边界的周围体积捆绑起来。此外,我们用光谱测试来描述距离一个拉蒂点的平均值。作为一个应用,我们推想,光谱测试 -- -- 连同异位差异 -- -- 对为索博列夫功能近端设定的拉蒂点是否合适至关重要。