On the Small Ball Inequality in All Dimensions

By D. Bilyk, Michael T. Lacey, A. Vagharshakyan. 2007

126 citations recorded by the source corpus.

Let h_R denote an L ^{\infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{\infty} norm of the `hyperbolic' sums $$ \sum _{|R|=2 ^{-n}} \alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{\eta} for some positive \eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b5375389f60979958608fa6348c9092442c41fee

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Numerical Analysis

Citations in the corpus

Hyperbolic Cross Approximation (2016, methodology): “Recent results on the Small Ball Inequality for the Haar system (see [38], [39] and the remark after (2.”
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration (2010, background): “This breakthrough led then to the paper of Bilyk, Lacey & Vagharshakyan [16] where it is shown that for any s ∈ N, s ≥ 2, there is a cs > 0 and a 0 < ηs < 1 2 with the property that for any point set P consisting of N points in the s-dimensional unit cube we have D∗ N (P) ≥ cs (log N)(s−1)/2+ηs N .”
Discrepancy Theory and Quasi-Monte Carlo Integration (2014, unknown): “For s ≥ 3 the lower bound on κs has recently been improved to κs ≥ (s − 1)/2 + δs for some unknown 0 < δs < 1/2; see Bilyk, Lacey and Vagharshakyan [8].”