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15 May 2008 On the small ball inequality in three dimensions
Dmitriy Bilyk, Michael T. Lacey
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Duke Math. J. 143(1): 81-115 (15 May 2008). DOI: 10.1215/00127094-2008-016

Abstract

Let hR denote an L-normalized Haar function adapted to a dyadic rectangle R[0,1]3. We show that there is a positive η<1/2 so that for all integers n and coefficients α(R), we have 2-n|R|=2-n|α(R)|n1-η|R|=2-nα(R)hR. This is an improvement over the trivial estimate by an amount of n-η, while the small ball conjecture says that the inequality should hold with η=1/2. There is a corresponding lower bound on the L-norm of the discrepancy function of an arbitrary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension three, is that of József Beck [1, Theorem 1.2], in which the improvement over the trivial estimate was logarithmic in n. We find several simplifications and extensions of Beck's argument to prove the result above

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Dmitriy Bilyk. Michael T. Lacey. "On the small ball inequality in three dimensions." Duke Math. J. 143 (1) 81 - 115, 15 May 2008. https://doi.org/10.1215/00127094-2008-016

Information

Published: 15 May 2008
First available in Project Euclid: 23 May 2008

zbMATH: 1202.42007
MathSciNet: MR2414745
Digital Object Identifier: 10.1215/00127094-2008-016

Subjects:
Primary: 11K38, 42A05
Secondary: 41A46, 42A55, 60E15, 60G17, 60J65

Rights: Copyright © 2008 Duke University Press

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Vol.143 • No. 1 • 15 May 2008
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