Kamarul Haili, Hailiza
(2006)
*A metric discrepancy estimate for a real sequence.*
Matematika, 22
(1).
pp. 25-30.
ISSN 0127-8274

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## Abstract

A general metrical result of discrepancy estimate related to uniform distribution is proved in this paper. It has been proven by J.W.S Cassel and P.Erdos \& Koksma in [2] under a general hypothesis of $(g_n (x))_{n = 1}^\infty$ that for every $\varepsilon > 0$, $$D(N,x) = O(N^{\frac{{ - 1}}{2}} (\log N)^{\frac{5}{2} + \varepsilon } )$$ for almost all $x$ with respect to Lebesgue measure. This discrepancy estimate was improved by R.C. Baker [5] who showed that the exponent $\frac{5}{2} + \varepsilon$ can be reduced to $\frac{3}{2} + \varepsilon$ in a special case where $g_n (x) = a_n x$ for a sequence of integers $(a_n )_{n = 1}^\infty$. This paper extends this result to the case where the sequence $(a_n )_{n = 1}^\infty$ can be assumed to be real. The lighter version of this theorem is also shown in this paper.

Item Type: | Article |
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Subjects: | Q Science > QA Mathematics |

ID Code: | 60 |

Deposited By: | Mohd Fuad Mohd Yusof |

Deposited On: | 12 Aug 2015 03:24 |

Last Modified: | 13 Aug 2015 04:36 |

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