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

## TL;DR

This paper establishes bounds on the isotropic discrepancy of lattice point sets using the spectral test, demonstrating that the discrepancy's order of magnitude is optimal and proportional to N^{-1/d} in dimension d.

## Contribution

It connects isotropic discrepancy bounds directly with the spectral test, providing optimal order estimates for lattice point sets in any dimension.

## Key findings

- Isotropic discrepancy can be bounded by spectral test values.
- The minimal order of isotropic discrepancy is N^{-1/d} for lattice sets.
- This order is proven to be optimal in dimension d.

## 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 $d$.