Optimal and typical \(L^2\) discrepancy of 2-dimensional lattices

We undertake a detailed study of the \(L^2\) $L^2$ 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 \(L^2\) $L^2$ 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 \(L^2\) $L^2$ discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.