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The exponent of discrepancy is at most

Authors:	Grzegorz W Wasilkowski Henryk Wozniakowski

Institution:	Department of Computer Science, University of Kentucky, Lexington, Kentucky 40506 ; Department of Computer Science, Columbia University, New York, New York 10027 and Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Abstract:	We study discrepancy with arbitrary weights in the norm over the -dimensional unit cube. The exponent of discrepancy is defined as the smallest for which there exists a positive number such that for all and all $\varepsilon \le 1$ there exist $K\varepsilon ^{-p}$ points with discrepancy at most $\varepsilon$ . It is well known that $p^\in (1,2]$ . We improve the upper bound by showing that $\begin{displaymath}p^\le 1.4778842.\end{displaymath}$ This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent is .

Keywords:	Discrepancy multivariate integration average case

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