Lp-discrepancy and beyond of higher order digital sequences

Description

The Lp-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 Lp-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 L2-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 Lp-discrepancy simultaneously for all p 2 (1;1). 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. The talk is based on joint work with Josef Dick, Lev Markhasin, and Friedrich Pillichshammer.

Period	16 Aug 2016
Event title	MCQMC 2016, 12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Stanford, California, August 14-19, 2016
Event type	Conference
Location	United StatesShow on map