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 $d ge 3$. In particular, we use dyadic harmonic analysis to prove that for the so-called digital nets of order $2$ the BMO${}^d$ and $exp big( L^{2/(d-1)} big)$ norms of the discrepancy function are bounded above by $(log N)^{frac{d-1}{2}}$. 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 $L_infty$ asymptotics of the discrepancy function in higher dimensions, which is still elusive.