General Discrepancy Estimates III: The Erdös-Turán-Koksma Inequality for the Haar Function System.
General discrepancy estimates: the Walsh function system
Gleichverteilung und die willkürlichen Funktionen von Poincaré
Improved discrepancy bounds for hybrid sequences involving Halton sequences
Improved upper bounds for the star discrepancy of digital nets in dimension 3
Improvements of the discrepancy of the van der Corput sequence.
Improvements on the star discrepancy of (t,s)-sequences
Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number generators
Irreducible Sobol' sequences in prime power bases
Sobol' sequences are a popular family of low-discrepancy sequences, in spite of requiring primitive polynomials instead of irreducible ones in later constructions by Niederreiter and Tezuka. We introduce a generalization of Sobol' sequences that removes this shortcoming and that we believe has the potential of becoming useful for practical applications. Indeed, these sequences preserve two important properties of the original construction proposed by Sobol': their generating matrices are non-singular...
Irregularities for distribution IX
Irregularities of continuous distributions
This paper deals with a continuous analogon to irregularities of point distributions. If a continuous fonction where is a compact body, is interpreted as a particle’s movement in time, then the discrepancy measures the difference between the particle’s stay in a proper subset and the volume of the subset. The essential part of this paper is to give lower bounds for the discrepancy in terms of the arc length of , . Furthermore it is shown that these estimates are the best possible despite of...
Irregularities of Distribution. IV.
Irregularities of distribution of digital -sequences in prime base.
Irregularities of distribution. VI
Irregularities of Point Distributions Relative to Homothetic Convex Bodies I.
Kronecker-type sequences and nonarchimedean diophantine approximations
discrepancy of generalized Zaremba point sets
We give an exact formula for the discrepancy of a class of generalized two-dimensional Hammersley point sets in base , namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base . The Zaremba point set introduced by White in 1975 is the special case where the shifts are taken repeatedly in sequential order, hence needing at least points to obtain the optimal order of discrepancy. On the...
- and -discrepancy of (order 2) digital nets
Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the -discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed...
-discrepancy and statistical independence of sequences
We characterize statistical independence of sequences by the -discrepancy and the Wiener -discrepancy. Furthermore, we find asymptotic information on the distribution of the -discrepancy of sequences.
L₂ discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations