The L2 Discrepancy of Irrational Lattices

Dmitriy Bilyk

doi:10.1007/978-3-642-41095-6_11

The L² Discrepancy of Irrational Lattices

Dmitriy Bilyk

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

6 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 language	English (US)
Title of host publication	Monte Carlo and Quasi-Monte Carlo Methods 2012
Pages	289-296
Number of pages	8
DOIs	https://doi.org/10.1007/978-3-642-41095-6_11
State	Published - 2013
Event	10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012 - Sydney, NSW, Australia Duration: Feb 13 2012 → Feb 17 2012

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	65
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012
Country/Territory	Australia
City	Sydney, NSW
Period	2/13/12 → 2/17/12

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10.1007/978-3-642-41095-6_11

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Bilyk, D 2013, The L² Discrepancy of Irrational Lattices. in Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics and Statistics, vol. 65, pp. 289-296, 10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012, Sydney, NSW, Australia, 2/13/12. https://doi.org/10.1007/978-3-642-41095-6_11

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