This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12]. 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.

本文是 Pillichshammer 和 Sonnleitner (2020) [12] 近期论文的后续。我们表明格点集的各向同性差异至多是$d{2}^{2(d+1)}$倍其谱测试，从而纠正对维度d的依赖以及上述论文定理 2 中上界证明的不准确性。主要任务是对包含在单位立方体中的凸集边界的邻域体积进行界定。此外，我们根据光谱测试表征到格点集的距离的平均值。作为一个应用，我们推断光谱测试——以及随之而来的各向同性差异——对于用于近似 Sobolev 函数的格点集的适用性至关重要。