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

Dmitriy Bilyk, Lev Markhasin

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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.

Original languageEnglish (US)
Pages (from-to)249-269
Number of pages21
JournalJournal d'Analyse Mathematique
Volume135
Issue number1
DOIs
StatePublished - Jun 1 2018

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