## Abstract

Let x_{0}, x_{1}, ... be a sequence of points in [0, 1)^{s}. A subset S of [0, 1)^{s} is called a bounded remainder set if there exist two real numbers a and C such that, for every integer N, |card{n < N | x_{n} ∈ S} − aN| < C. Let (x_{n})_{n}≥0 be an s−dimensional Halton-type sequence obtained from a global function field, b ≥ 2, γ = (γ_{1}, ..., γ_{s}), γ_{i} ∈ [0, 1), with b-adic expansion γi = γi,_{1}b^{−} ^{1} + γi,_{2}b^{−} ^{2} + ..., i = 1, ..., s. In this paper, we prove that [0, γ_{1}) × ... × [0, γ_{s}) is the bounded remainder set with respect to the sequence (x_{n})_{n}≥0 if and only if _{1}max _{≤}i≤_{s} max{j ≥ 1 | γ_{i,j} ≠ 0} < ∞. We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.

Translated title of the contribution | On a bounded remainder set for (t,s) sequences I |
---|---|

Original language | English |

Pages (from-to) | 222-245 |

Number of pages | 24 |

Journal | Chebyshevskii Sbornik |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Publisher Copyright:© 2019 State Lev Tolstoy Pedagogical University. All rights reserved.

## Keywords

- (t
- Bounded remainder set
- Halton type sequences
- S) sequence