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 $\exp \left(L^{2\mathrm{/}(d-1)}\right)$ 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 $L_{\mathrm{\infty }}$ asymptotics of the discrepancy function in higher dimensions, which is still elusive.