BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension
This provides the first tight bounds in BMO and exponential Orlicz spaces for high-dimensional discrepancy, an important step toward resolving the L_∞ asymptotics.
The paper proves that for digital nets of order 2 in dimension d≥3, the BMO and exponential Orlicz norms of the discrepancy function are bounded by (log N)^{(d-1)/2}, confirming a recent conjecture and bridging L_p bounds with the open L_∞ problem.
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.