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

2014/11/21 by Bilyk, Dmitriy, Markhasin, Lev
#Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1411.5794

Abstract

In the current paper we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension d ≥ 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 ( L^2/(d-1) ) norms of the discrepancy function are bounded above by (log N)^(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_∞ asymptotics of the discrepancy function in higher dimensions, which is still elusive.

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