Optimal and typical L² discrepancy of 2-dimensional lattices

We undertake a detailed study of the L² L 2 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 L² 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 L² L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.