Optimal and typical $L^2$ discrepancy of 2-dimensional lattices

TL;DR

This paper provides a comprehensive analysis of the $L^2$ discrepancy in 2D lattices, characterizing optimal cases via continued fractions and exploring asymptotics for typical irrationals and random lattices.

Contribution

It offers a full characterization of lattices with optimal $L^2$ discrepancy using continued fraction partial quotients and computes precise asymptotics for specific classes of irrationals.

Findings

Lattices with optimal $L^2$ discrepancy are characterized by their continued fraction expansion.

Asymptotics of $L^2$ discrepancy are derived for almost every irrational lattice.

Limit distributions are established for randomly selected rational and irrational lattices.

Abstract

We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ 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 $L^2$ discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.