The Exponent of Discrepancy Is at Least 1.0669

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Abstract:

LetP⊂[0, 1]dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑zP w(z)=1. TheL2-discrepancy of the weighted set (P, w) is defined as theL2-average ofD(x)=vol(Bx)-w(PBx) overx∈[0, 1]d, where vol(Bx) is the volume of thed-dimensional intervalBx=∏dk=1 [0, xk). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd1 and all>0 there exists a weighted set of at mostK-ppoints in [0, 1]dwithL2-discrepancy at most, whereK=K(p) is a suitable number independent ofandd. Wasilkowski and Woz´niakowski proved thatp*1.4779, by combining known bounds for the error of numerical integration and using their relation toL2-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*1.04882, and by a slight modification of the proof we getp*1.0669. Determiningp* exactly seems to be quite a difficult problem.

Document Type: Research Article

Affiliations: Department of Applied Mathematics, Charles University, Malostranské nám. 25, Prague 1, 118 00, Czech Republic

Publication date: December 1, 1998

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