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L 2 discrepancy of generalized Zaremba point sets

Henri Faure, Friedrich 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...

L p - and S p , q r B -discrepancy of (order 2) digital nets

Lev Markhasin (2015)

Acta Arithmetica

Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the L p -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...

L p -discrepancy and statistical independence of sequences

Peter J. Grabner, Oto Strauch, Robert Franz Tichy (1999)

Czechoslovak Mathematical Journal

We characterize statistical independence of sequences by the L p -discrepancy and the Wiener L p -discrepancy. Furthermore, we find asymptotic information on the distribution of the L 2 -discrepancy of sequences.

Low-discrepancy point sets for non-uniform measures

Christoph Aistleitner, Josef Dick (2014)

Acta Arithmetica

We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on [ 0 , 1 ] d there exists a point set x 1 , . . . , x N [ 0 , 1 ] d whose star-discrepancy with respect to μ is of order D N * ( x 1 , . . . , x N ; μ ) ( ( l o g N ) ( 3 d + 1 ) / 2 ) / N . For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy...

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