## 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 L^{2} 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) |
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Title of host publication | Monte Carlo and Quasi-Monte Carlo Methods 2012 |

Pages | 289-296 |

Number of pages | 8 |

DOIs | |

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 |
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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 |
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Country/Territory | Australia |

City | Sydney, NSW |

Period | 2/13/12 → 2/17/12 |

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