Optimal $L_p$-discrepancy bounds for second order digital sequences

Abstract

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 $p\in (1,\infty )$.

A journal requested to split the paper into two parts. This is the first part which contains the results on the $L_p$ discrepancy

References in corpus (3)

• From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
• Discrepancy and integration in function spaces with dominating mixed smoothness
• L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets