<abstract xmlns="http://www.w3.org/1999/xhtml">It is known that there is a constant <italic>c</italic> > 0 such that for every sequence <italic>x</italic>1, <italic>x</italic>2, . . . in [0, 1) we have for the star discrepancy <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0001_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>N</mi><mo>*</mo></msubsup></math><tex-math>$D_N^* $</tex-math></alternatives></inline-formula> of the first <italic>N</italic> elements of the sequence that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0001_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ND</mi><mi>N</mi><mo>*</mo></msubsup><mo>≥</mo><mi>c</mi><mo>⋅</mo><mi>log</mi><mo>⁡</mo><mi>N</mi></math><tex-math>$ND_N^* \ge c \cdot \log N$</tex-math></alternatives></inline-formula> holds for infinitely many <italic>N</italic>. Let <italic>c</italic>∗ be the supremum of all such <italic>c</italic> with this property. We show <italic>c</italic>∗ > 0.065664679 . . . , thereby slightly improving the estimates known until now.</abstract>

It is known that there is a constant c > 0 such that for every sequence x1, x2, . . . in [0, 1) we have for the star discrepancy DN*$D_N^* $ of the first N elements of the sequence that NDN*≥c⋅log⁡N$ND_N^* \ge c \cdot \log N$ holds for infinitely many N. Let c∗ be the supremum of all such c with this property. We show c∗ > 0.065664679 . . . , thereby slightly improving the estimates known until now.

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

Institute of Financial Mathematics and Applied Number Theory, University Linz, Altenbergerstraße 69, 4040 Linz,

Uniform distribution theory

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{"article-title":"An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval"}

It is known that there is a constant c > 0 such that for every sequence x1, x2, . . . in [0, 1) we have for the star discrepancy DN*$D_N^* $ of the first N...

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

Online veröffentlicht: 13. Jan. 2017

Seitenbereich: 1 - 14

Eingereicht: 01. Dez. 2014

Akzeptiert: 02. Okt. 2015

DOI: https://doi.org/10.1515/udt-2016-0001

Schlüsselwörter
uniform distribution, discrepancy

© 2016 Gerhard Larcher et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

Online veröffentlicht: 13. Jan. 2017

Seitenbereich: 1 - 14

Eingereicht: 01. Dez. 2014

Akzeptiert: 02. Okt. 2015

DOI: https://doi.org/10.1515/udt-2016-0001

Schlüsselwörteruniform distribution, discrepancy

© 2016 Gerhard Larcher et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Schlüsselwörter
uniform distribution, discrepancy