$L_p$-discrepancy and beyond of higher order digital sequences

Josef Dick, Aicke Hinrichs, Lev Markhasin, Friedrich Pillichshammer

Published 2016-01-27Version 1

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all $p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in the $d$-dimensional unit cube, but it remained an open question whether this lower bound is best possible in the order of magnitude until recently. In 2014 Dick and Pillichshammer gave a first construction of an infinite sequence whose order of $L_2$-discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite $p > 1$. We consider so-called order $2$ digital $(t,d)$-sequences over the finite field with two elements and show that such sequences achieve the optimal order of $L_p$-discrepancy simultaneously for all $p \in (1,\infty)$. Beyond this result, we estimate the norm of the discrepancy function of those sequences also in the space of bounded mean oscillation, exponential Orlicz spaces, Besov and Triebel-Lizorkin spaces and give some corresponding lower bounds which show that the obtained upper bounds are optimal in the order of magnitude.