The L2 Discrepancy of Two-Dimensional Lattices

Abstract

Let α be an irrational number with bounded partial quotients of the continued fraction ak. It is well known that symmetrizations of the irrational lattice have optimal order of L2 discrepancy, √log N. The same is true for their rational approximations, where pn/qn is the nth convergent of α. However, the question whether and when the symmetrization is really necessary remained wide open. We show that the L2 discrepancy of the nonsymmetrized lattice Ln(α) grows as in particular, characterizing the lattices for which the L2 discrepancy is optimal. © Springer Science+Business Media, LLC 2013.