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

Abstract

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.