On the Small Ball Inequality in Three Dimensions
| dc.creator | Lacey, Michael T | |
| dc.creator | Bilyk, Dmitry | |
| dc.date | 2006-09-28 | |
| dc.date | 2007-06-21 | |
| dc.date.accessioned | 2026-07-07T08:11:26Z | |
| dc.date.available | 2026-07-07T08:11:26Z | |
| dc.description | 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 $η$ so that for all integers $n$, and coefficients $ α(R)$ we have 2 ^{-n} \sum_{\abs{R}=2 ^{-n}} \abs{α(R)} {}\lesssim{} n ^{1 - η} \NOrm \sum_{\abs{R}=2 ^{-n}} α(R) h_R >.\infty . 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 $L ^{\infty}$ 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{ó}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. | |
| dc.description | 30 pages. Final version of the paper. To appear in Duke Math J | |
| dc.identifier | https://arxiv.org/abs/math/0609815 | |
| dc.identifier | http://arxiv.org/abs/math/0609815 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132120 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | On the Small Ball Inequality in Three Dimensions | |
| dc.type | text |