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|a (JST)26730892
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|a eng
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|a KRITZINGER, Ralph
|e verfasserin
|4 aut
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|a Digital nets in dimension two with the optimal order of Lp discrepancy
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|c 2019
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|a Text
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|a Nous étudions la discrépance Lp (p ϵ [1, ∞)) de réseaux digitaux de dimension 2. En 2001, Larcher et Pillichshammer ont identifié une classe de (0, n, 2)-réseaux pour lesquels la version symétrisée au sens de Davenport a une discrépance L₂ d’ordre log N / N , qui est optimal grâce au résultat célèbre de Roth. Cependant la question de savoir si la même borne s’applique à la discrépance des réseaux originaux est restée ouverte. Dans cet article, nous identifions les réseaux digitaux de la classe susmentionnée pour lesquels la symétrisation n’est pas nécessaire pour obtenir l’ordre optimal de la discrépance Lp pour p ϵ [1, ∞). Ce résultat est dans l’esprit d’un article de Bilyk de 2013, qui a étudié la discrépance L₂ des ensembles des points de la forme (κ/N, {κα}) pourκ = 0, 1, . . . ,N – 1 et a donné des propriétés diophantiennes de α qui garantissent l’ordre optimal de la discrépance L₂. We study the Lp discrepancy of two-dimensional digital nets for finite p. In the year 2001 Larcher and Pillichshammer identified a class of digital nets for which the symmetrized version in the sense of Davenport has L₂ discrepancy of the order log N / N , which is best possible due to the celebrated result of Roth. However, it remained open whether this discrepancy bound also holds for the original digital nets without any modification. In the present paper we identify nets from the above mentioned class for which the symmetrization is not necessary in order to achieve the optimal order of Lp discrepancy for all p ϵ [1, ∞). Our findings are in the spirit of a paper by Bilyk from 2013, who considered the L₂ discrepancy of lattices consisting of the elements (κ/N, {κα}) forκ = 0, 1, . . . ,N – 1, and who gave Diophantine properties of α which guarantee the optimal order of L₂ discrepancy.
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|a © Société Arithmétique de Bordeaux, 2019
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|a Mathematics
|x Pure mathematics
|x Algebra
|x Polynomials
|x Dyadics
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|a Mathematics
|x Pure mathematics
|x Algebra
|x Coefficients
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| 650 |
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|a Mathematics
|x Applied mathematics
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| 650 |
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|a Philosophy
|x Logic
|x Logical topics
|x Formal logic
|x Mathematical logic
|x Mathematical set theory
|x Lattice theory
|x Mathematical lattices
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| 650 |
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|a Mathematics
|x Mathematical objects
|x Mathematical intervals
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|a Mathematics
|x Mathematical expressions
|x Mathematical functions
|x Indicator functions
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|a Mathematics
|x Pure mathematics
|x Discrete mathematics
|x Number theory
|x Numbers
|x Real numbers
|x Rational numbers
|x Integers
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|a Mathematics
|x Mathematical objects
|x Mathematical sequences
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|a Mathematics
|x Pure mathematics
|x Discrete mathematics
|x Number theory
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|a research-article
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|a PILLICHSHAMMER, Friedrich
|e verfasserin
|4 aut
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|i Enthalten in
|t Journal de Théorie des Nombres de Bordeaux
|d Société Arithmétique de Bordeaux, 1993
|g 31(2019), 1, Seite 179-204
|w (DE-627)320967603
|w (DE-600)2028468-8
|x 21188572
|7 nnns
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|g volume:31
|g year:2019
|g number:1
|g pages:179-204
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|u https://www.jstor.org/stable/26730892
|3 Volltext
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|a AR
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|d 31
|j 2019
|e 1
|h 179-204
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