## 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 |
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Title of host publication | Monte Carlo and Quasi-Monte Carlo Methods - MCQMC 2016 |

Publisher | Springer New York LLC |

Pages | 169-180 |

Number of pages | 12 |

Volume | 241 |

ISBN (Print) | 9783319914350 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

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

### Conference

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

City | Stanford |

Period | 14/08/16 → 19/08/16 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)