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

TL;DR

This paper establishes new bounds for the discrepancy function in exponential Orlicz and BMO spaces for digital nets in arbitrary dimensions, advancing understanding of discrepancy estimates beyond traditional Lp bounds.

Contribution

It introduces novel discrepancy bounds in exponential Orlicz and BMO spaces for digital nets, bridging the gap between Lp bounds and the challenging L-infinity problem.

Findings

BMO^d and exponential Orlicz norms are bounded by (log N)^{(d-1)/2} for digital nets

The bounds support conjectures and align with known low-discrepancy constructions

Results provide an intermediate step towards understanding L-infinity discrepancy asymptotics

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.