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Dyadic diaphony of digital sequences

Friedrich Pillichshammer — 2007

Journal de Théorie des Nombres de Bordeaux

The dyadic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we give formulae for the dyadic diaphony of digital ( 0 , s ) -sequences over 2 , s = 1 , 2 . These formulae show that for fixed s { 1 , 2 } , the dyadic diaphony has the same values for any digital ( 0 , s ) -sequence. For s = 1 , it follows that the dyadic diaphony and the diaphony of special digital ( 0 , 1 ) -sequences are up to a constant the same. We give the exact asymptotic order of the dyadic diaphony of digital...

A generalization of NUT digital (0,1)-sequences and best possible lower bounds for star discrepancy

Henri FaureFriedrich Pillichshammer — 2013

Acta Arithmetica

In uniform distribution theory, discrepancy is a quantitative measure for the irregularity of distribution of a sequence modulo one. At the moment the concept of digital (t,s)-sequences as introduced by Niederreiter provides the most powerful constructions of s-dimensional sequences with low discrepancy. In one dimension, recently Faure proved exact formulas for different notions of discrepancy for the subclass of NUT digital (0,1)-sequences. It is the aim of this paper to generalize the concept...

L 2 discrepancy of generalized Zaremba point sets

Henri FaureFriedrich Pillichshammer — 2011

Journal de Théorie des Nombres de Bordeaux

We give an exact formula for the L 2 discrepancy of a class of generalized two-dimensional Hammersley point sets in base b , namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base b . The Zaremba point set introduced by White in 1975 is the special case where the b shifts are taken repeatedly in sequential order, hence needing at least b b points to obtain the optimal order of L 2 discrepancy. On the...

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