A note on isotropic discrepancy and spectral test of lattice point sets

Friedrich Pillichshammer; Mathias Sonnleitner

doi:10.1016/j.jco.2019.101441

paper

A note on isotropic discrepancy and spectral test of lattice point sets

arXiv:1907.06435 · doi:10.1016/j.jco.2019.101441

Abstract

We show that the isotropic discrepancy of a lattice point set can be bounded from below and from above in terms of the spectral test of the corresponding integration lattice. From this we deduce that the isotropic discrepancy of any $N$ -element lattice point set in $[0, 1)^{d}$ is at least of order $N^{- 1/ d}$ . This order of magnitude is best possible for lattice point sets in dimension $d$ .