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.
|Original language||English (US)|
|Title of host publication||Recent Advances in Harmonic Analysis and Applications|
|Subtitle of host publication||In Honor of Konstantin Oskolkov|
|Publisher||Springer New York LLC|
|Number of pages||15|
|State||Published - 2013|
|Name||Springer Proceedings in Mathematics and Statistics|
Bibliographical noteFunding Information:
The research of the authors is sponsored by NSF grants DMS 1101519, DMS 0906260, and EAR 0934747.