The L ² Discrepancy of Irrational Lattices

Abstract

It is well known that, when α has bounded partial quotients, the lattices $\big\{\big(k/N,\{k\alpha \}\big)\big\}_{k=0}^{N-1}$ have optimal extreme discrepancy. The situation with the L ² discrepancy, however, is more delicate. In 1956 Davenport established that a symmetrized version of this lattice has L ² discrepancy of the order $\sqrt{\log N}$, which is the lowest possible due to the celebrated result of Roth. However, it remained unclear whether this holds for the original lattices without any modifications. It turns out that the L ² discrepancy of the lattice depends on much finer Diophantine properties of α, namely, the alternating sums of the partial quotients. In this paper we extend the prior work to arbitrary values of α and N. We heavily rely on Beck’s study of the behavior of the sums $\sum \big(\{k\alpha \} -\frac{1} {2}\big)$.