TY - JOUR

T1 - On the small ball inequality in three dimensions

AU - Bilyk, Dmitriy

AU - Lacey, Michael T.

PY - 2008/5/15

Y1 - 2008/5/15

N2 - 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 (Equation Presented) 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.

AB - 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 (Equation Presented) 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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U2 - 10.1215/00127094-2008-016

DO - 10.1215/00127094-2008-016

M3 - Article

AN - SCOPUS:44949144902

SN - 0012-7094

VL - 143

SP - 81

EP - 115

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 1

ER -