On the small ball inequality in three dimensions

By Michael T. Lacey, D. Bilyk. 2006

74 citations recorded by the source corpus.

We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a dyadic rectangle $R\subset [0,1] ^{3}$. We show that there is a postive $\eta$ so that for all integers $n$, and coefficients $ \alpha (R)$ we have 2 ^{-n} \sum_{\abs{R}=2 ^{-n}} \abs{\alpha(R)} {}\lesssim{} n ^{1 - \eta} \NOrm \sum_{\abs{R}=2 ^{-n}} \alpha(R) h_R >.\infty . This is an improvement over the `trivial' estimate by an amount of $n ^{- \eta}$, and the optimal value of $\eta$ (which we do not prove) would be $ \eta =\frac12$. There is a corresponding lower bound on the $L ^{\infty}$ norm of the Discrepancy function of an arbitary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension 3, is that of J{o}zsef Beck \cite{MR1032337}, 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7978697b4a8f62f194dafc37b1cd3c391ea52409

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In Algebra and Number Theory

In Applied Mathematics

In Numerical Analysis

Citations in the corpus

Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration (2010, methodology): “An improvement of Becks result in dimension s = 3 has been shown by Bilyk & Lacey [15].”
Hyperbolic Cross Approximation (2016, methodology): “Recent results on the Small Ball Inequality for the Haar system (see [38], [39]) allow us to improve a trivial lower bound to the following one for r = 1 and all p <∞, d ≥ 3: dm(W 1 p, L∞) & m −1(logm)d−1+δ(d), δ(d) > 0.”