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

TL;DR

This paper refines the relationship between isotropic discrepancy and the spectral test for lattice point sets, correcting previous inaccuracies and highlighting their importance in Sobolev space approximation.

Contribution

It establishes a corrected upper bound linking isotropic discrepancy and the spectral test, and characterizes their role in Sobolev function approximation.

Findings

Isotropic discrepancy is at most a dimension-dependent multiple of the spectral test.

The spectral test is essential for the effectiveness of lattice point sets in Sobolev approximation.

The paper corrects previous bounds and provides new insights into lattice discrepancy measures.

Abstract

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.