A metric discrepancy estimate for a real sequence
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 $\varepsilon > 0$, $$D(N,x) = O(N^{\frac{{ - 1}}...
Saved in:
Main Author: | |
---|---|
Format: | Article |
Language: | English |
Published: |
2006
|
Subjects: | |
Online Access: | http://eprints.utm.my/id/eprint/60/1/A_Metric_Discrepancy_Estimate_for_A_Real_Sequence.pdf http://eprints.utm.my/id/eprint/60/ http://161.139.72.2/oldfs/images/stories/matematika/20062213.pdf |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
id |
my.utm.60 |
---|---|
record_format |
eprints |
spelling |
my.utm.602015-08-13T04:36:48Z http://eprints.utm.my/id/eprint/60/ A metric discrepancy estimate for a real sequence Kamarul Haili, Hailiza QA Mathematics A general metrical result of discrepancy estimate related to uniform distribution is proved in this paper. It has been proven by J.W.S Cassel and P.Erdos \& Koksma in [2] under a general hypothesis of $(g_n (x))_{n = 1}^\infty$ that for every $\varepsilon > 0$, $$D(N,x) = O(N^{\frac{{ - 1}}{2}} (\log N)^{\frac{5}{2} + \varepsilon } )$$ for almost all $x$ with respect to Lebesgue measure. This discrepancy estimate was improved by R.C. Baker [5] who showed that the exponent $\frac{5}{2} + \varepsilon$ can be reduced to $\frac{3}{2} + \varepsilon$ in a special case where $g_n (x) = a_n x$ for a sequence of integers $(a_n )_{n = 1}^\infty$. This paper extends this result to the case where the sequence $(a_n )_{n = 1}^\infty$ can be assumed to be real. The lighter version of this theorem is also shown in this paper. 2006-01 Article NonPeerReviewed application/pdf en http://eprints.utm.my/id/eprint/60/1/A_Metric_Discrepancy_Estimate_for_A_Real_Sequence.pdf Kamarul Haili, Hailiza (2006) A metric discrepancy estimate for a real sequence. Matematika, 22 (1). pp. 25-30. ISSN 0127-8274 http://161.139.72.2/oldfs/images/stories/matematika/20062213.pdf |
institution |
Universiti Teknologi Malaysia |
building |
UTM Library |
collection |
Institutional Repository |
continent |
Asia |
country |
Malaysia |
content_provider |
Universiti Teknologi Malaysia |
content_source |
UTM Institutional Repository |
url_provider |
http://eprints.utm.my/ |
language |
English |
topic |
QA Mathematics |
spellingShingle |
QA Mathematics Kamarul Haili, Hailiza A metric discrepancy estimate for a real sequence |
description |
A general metrical result of discrepancy estimate related to uniform distribution is proved in this paper. It has been proven by J.W.S Cassel and P.Erdos \& Koksma in [2] under a general hypothesis of $(g_n (x))_{n = 1}^\infty$ that for every $\varepsilon > 0$,
$$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. |
format |
Article |
author |
Kamarul Haili, Hailiza |
author_facet |
Kamarul Haili, Hailiza |
author_sort |
Kamarul Haili, Hailiza |
title |
A metric discrepancy estimate for a real sequence |
title_short |
A metric discrepancy estimate for a real sequence |
title_full |
A metric discrepancy estimate for a real sequence |
title_fullStr |
A metric discrepancy estimate for a real sequence |
title_full_unstemmed |
A metric discrepancy estimate for a real sequence |
title_sort |
metric discrepancy estimate for a real sequence |
publishDate |
2006 |
url |
http://eprints.utm.my/id/eprint/60/1/A_Metric_Discrepancy_Estimate_for_A_Real_Sequence.pdf http://eprints.utm.my/id/eprint/60/ http://161.139.72.2/oldfs/images/stories/matematika/20062213.pdf |
_version_ |
1643643023809052672 |
score |
13.18916 |