The L2 Discrepancy of Irrational Lattices

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

It is well known that, when α has bounded partial quotients, the lattices{(k/N,{kα}) }N-1 k=0 have optimal extreme discrepancy. The situation with the L2 discrepancy, however, is more delicate. In 1956 Davenport established that a symmetrized version of this lattice has L2discrepancy of the orderf p √logN, which is the lowest possible due to the celebrated result of Roth. However, it remained unclear whether this holds for the original lattices without anymodifications. It turns out that the L2discrepancy 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 Σ({kα}-1/2.

Original languageEnglish (US)
Title of host publicationMonte Carlo and Quasi-Monte Carlo Methods 2012
Pages289-296
Number of pages8
DOIs
StatePublished - 2013
Event10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012 - Sydney, NSW, Australia
Duration: Feb 13 2012Feb 17 2012

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume65
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

Other10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012
Country/TerritoryAustralia
CitySydney, NSW
Period2/13/122/17/12

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