### Abstract

It is well-known that for every N≥ 1 and d≥ 1 there exist point sets x_{1}, ⋯, x_{N}∈ [0, 1]^{d} whose discrepancy with respect to the Lebesgue measure is of order at most (log N)^{d} ^{-} ^{1}N^{-1}. In a more general setting, the first author proved together with Josef Dick that for any normalized measure μ on [0, 1 ]^{d} there exist points x_{1}, ⋯, x_{N} whose discrepancy with respect to μ is of order at most (log N)^{(} ^{3} ^{d} ^{+} ^{1} ^{)} ^{/} ^{2}N^{- 1}. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d-12N-1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.

Original language | English (US) |
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Title of host publication | Monte Carlo and Quasi-Monte Carlo Methods - MCQMC 2016 |

Editors | Peter W. Glynn, Art B. Owen |

Publisher | Springer New York LLC |

Pages | 169-180 |

Number of pages | 12 |

ISBN (Print) | 9783319914350 |

DOIs | |

State | Published - Jan 1 2018 |

Event | 12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2016 - Stanford, United States Duration: Aug 14 2016 → Aug 19 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 241 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | 12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2016 |
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Country | United States |

City | Stanford |

Period | 8/14/16 → 8/19/16 |

### Keywords

- Gates of Hell
- Low-discrepancy sequences
- Non-uniform sampling
- Tusnády’s problem
- combinatorial discrepancy

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## Cite this

*Monte Carlo and Quasi-Monte Carlo Methods - MCQMC 2016*(pp. 169-180). (Springer Proceedings in Mathematics and Statistics; Vol. 241). Springer New York LLC. https://doi.org/10.1007/978-3-319-91436-7_8