2006/09/28 by Michael T. Lacey, Michael T Lacey, Lacey, Michael T +2 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #math.CA
paper · pdf · doi:10.48550/arxiv.math/0609815
30 pages. Final version of the paper. To appear in Duke Math J
openalex publication_date 2006/09/28 · arxiv created 2007/06/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let hR denote an L ∞ normalized Haar function adapted to a dyadic rectangle R⊂ [0,1] 3. We show that there is a postive η so that for all integers n, and coefficients α(R) we have 2 -n ∑_\absR=2 -n \absα(R) \lesssim n 1 - η \NOrm ∑_\absR=2 -n α(R) hR >.∞ . This is an improvement over the `trivial' estimate by an amount of n - η, and the optimal value of η (which we do not prove) would be η=\frac12. There is a corresponding lower bound on the L ∞ 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 József Beck \citeMR1032337, 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.