### Abstract

Given a finite sequence U_{N} = {u1,., uN} of points contained in the d-dimensional unit torus, we consider the L^{2} discrepancy between the integral of a given function and the Riemann sums with respect to translations of U_{N}. We show that with positive probability, the L^{2} discrepancy of other sequences close to U_{N} in a certain sense preserves the order of decay of the discrepancy of U_{N}. We also study the role of the regularity of the given function.

Original language | English |
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Pages (from-to) | 3187-3196 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2009 |

### Bibliographical note

Copyright 2009 American Mathematical Society. First published in Proceedings of the American Mathematical Society, Volume 137, Issue 10, pp. 3187-3196, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-09-09975-4.## Fingerprint Dive into the research topics of 'Discrepancy for randomized riemann sums'. Together they form a unique fingerprint.

## Cite this

Brandolini, L., Chen, W., Gigante, G., & Travaglini, G. (2009). Discrepancy for randomized riemann sums.

*Proceedings of the American Mathematical Society*,*137*(10), 3187-3196. https://doi.org/10.1090/S0002-9939-09-09975-4