On the relation of the spectral test to isotropic discrepancy and $L_q$-approximation in Sobolev spaces

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.