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A Metric Discrepancy Estimate for A Real Sequence

Author(s): Hailiza Kamarul Haili

Journal: Matematika
ISSN 0127-8274

Volume: 22;
Issue: 1;
Start page: 25;
Date: 2006;
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Keywords: Discrepancy | uniform distribution | Lebesgue measure | almost everywhere.

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 (gn(x))¥n=1 that for every ε > 0,                     D(N, x) = O(N-½ (log N)5/2+ε)for almost all x with respect to Lebesgue measure. This discrepancy estimate was improved by R.C. Baker [5] who showed that the exponent 5/2+ε can be reduced to 3/2+ε in a special case where gn(x) = anx for a sequence of integers (an)¥n=1. This paper extends this result to the case where the sequence (an)¥n=1 can be assumed to be real. The lighter version of this theorem is also shown in this paper.
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