Abstract
Let denote an -normalized Haar function adapted to a dyadic rectangle . We show that there is a positive so that for all integers and coefficients , we have This is an improvement over the trivial estimate by an amount of , while the small ball conjecture says that the inequality should hold with . There is a corresponding lower bound on the -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 . We find several simplifications and extensions of Beck's argument to prove the result above
Citation
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
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