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 $\varepsilon > 0$, $$D(N,x) = O(N^{\frac{{ - 1}}...
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| Format: | Article |
| Language: | English |
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2006
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| Online Access: | http://eprints.utm.my/60/1/A_Metric_Discrepancy_Estimate_for_A_Real_Sequence.pdf |
| _version_ | 1796853029262065664 |
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| author | Kamarul Haili, Hailiza |
| author_facet | Kamarul Haili, Hailiza |
| author_sort | Kamarul Haili, Hailiza |
| collection | ePrints |
| description | A general metrical result of discrepancy estimate related to uniform distribution is proved in this paper. It has been proven by J.W.S Cassel and P.Erdos \& Koksma in [2] under a general hypothesis of $(g_n (x))_{n = 1}^\infty$ that for every $\varepsilon > 0$,
$$D(N,x) = O(N^{\frac{{ - 1}}{2}} (\log N)^{\frac{5}{2} + \varepsilon } )$$
for almost all $x$ with respect to Lebesgue measure. This discrepancy estimate was improved by R.C. Baker [5] who showed that the exponent $\frac{5}{2} + \varepsilon$ can be reduced to $\frac{3}{2} + \varepsilon$ in a special case where $g_n (x) = a_n x$ for a sequence of integers $(a_n )_{n = 1}^\infty$. This paper extends this result to the case where the sequence $(a_n )_{n = 1}^\infty$ can be assumed to be real. The lighter version of this theorem is also shown in this paper. |
| first_indexed | 2024-03-05T17:53:37Z |
| format | Article |
| id | utm.eprints-60 |
| institution | Universiti Teknologi Malaysia - ePrints |
| language | English |
| last_indexed | 2024-03-05T17:53:37Z |
| publishDate | 2006 |
| record_format | dspace |
| spelling | utm.eprints-602015-08-13T04:36:48Z http://eprints.utm.my/60/ A metric discrepancy estimate for a real sequence Kamarul Haili, Hailiza QA Mathematics A general metrical result of discrepancy estimate related to uniform distribution is proved in this paper. It has been proven by J.W.S Cassel and P.Erdos \& Koksma in [2] under a general hypothesis of $(g_n (x))_{n = 1}^\infty$ that for every $\varepsilon > 0$, $$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/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 |
| spellingShingle | QA Mathematics Kamarul Haili, Hailiza A metric discrepancy estimate for a real sequence |
| title | 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_short | A metric discrepancy estimate for a real sequence |
| title_sort | metric discrepancy estimate for a real sequence |
| topic | QA Mathematics |
| url | http://eprints.utm.my/60/1/A_Metric_Discrepancy_Estimate_for_A_Real_Sequence.pdf |
| work_keys_str_mv | AT kamarulhailihailiza ametricdiscrepancyestimateforarealsequence AT kamarulhailihailiza metricdiscrepancyestimateforarealsequence |