The L2 Discrepancy of Two-Dimensional Lattices

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 {( μ/N,{μα}N−1 μ=0 have optimal order of L2 discrepancy, √ logN. The same is true for their rational approximations Ln(α) = {( μ/qn,{μ pn/qn} )}qn−1 μ=0 , 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 ‖D(Ln(α),x)‖2 ≈ max { log 1 2 qn, ∣ ∣ ∣∣ ∣ n ∑ k=0 (−1)ak ∣ ∣ ∣∣ ∣ }