# Optimal and typical L2 discrepancy of 2-dimensional lattices

## TL;DR

This paper analyzes the L2 discrepancy of 2D lattices, focusing on optimal configurations and typical behavior for both rational and irrational cases.

## Contribution

The paper provides a full characterization of optimal lattices using continued fractions and derives asymptotics for specific irrational numbers.

## Key findings

- Optimal L2 discrepancy is characterized via continued fraction partial quotients.
- Asymptotics for L2 discrepancy are computed for quadratic irrationals and Euler’s number e.
- Limit distributions are derived for randomly chosen rational and irrational lattices.

## Abstract

We undertake a detailed study of the \documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}L2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal \documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}L2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler’s number e. In the metric theory, we find the asymptotics of the \documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}L2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.