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In the current paper, we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension  ≥ 3. In particular, we use dyadic harmonic analysis to prove that the dyadic product BMO and exp() norms of the discrepancy function of so-called digital nets of order two are bounded above by (log). 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  bounds and the notorious open problem of finding the precise  asymptotics of the discrepancy function in higher dimensions, which is still elusive.

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