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Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 249–269 | Cite as

BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

  • Dmitriy Bilyk
  • Lev Markhasin
Article
  • 10 Downloads

Abstract

In the current paper, we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension d ≥ 3. In particular, we use dyadic harmonic analysis to prove that the dyadic product BMO and exp(L2/(d−1)) norms of the discrepancy function of so-called digital nets of order two are bounded above by (logN)(d−1)/2. The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood Lp bounds and the notorious open problem of finding the precise L asymptotics of the discrepancy function in higher dimensions, which is still elusive.

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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Institut für Stochastik und AnwendungenUniversität StuttgartStuttgartGermany

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