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

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(L^2/(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 L_p bounds and the notorious open problem of finding the precise L_∞ asymptotics of the discrepancy function in higher dimensions, which is still elusive.