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Abstract. The discrepancy function measures the deviation of the empirical distribution of a point set in [0, 1] d from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel-Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences. §1. Background. Let d, N be positive integers and let P N ,d be a point set inHere χ A denotes the characteristic function of a subset A ∈ R d . Thus D P N ,d (x) denotes the difference between the actual and expected proportions of points that

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“…In the case d = 1 this result has been recently shown in [19,Theorem 1] which is based on the symmetrized van der Corput sequence.…”

Section: Besov and Triebel-lizorkin Spaces With Dominating Mixed Smoosupporting

confidence: 54%

Discrepancy of Second Order Digital Sequences in Function Spaces With Dominating Mixed Smoothness

Dick¹,

Hinrichs

Markhasin

et al. 2017

Mathematika

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“…A generalization of this result to van der Corput sequences in arbitrary base b ≥ 2 has been shown quite recently by Kritzinger [28]. See also the recent survey article [22] and the references therein for more information about symmetrized van der Corput sequences.…”

Section: Introductionmentioning

confidence: 59%

Optimal L p -discrepancy bounds for second order digital sequences

Dick¹,

Hinrichs²,

Markhasin³

et al. 2017

Isr. J. Math.

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The L p -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 L p -discrepancy of general infinite sequences in the ddimensional 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 L 2 -discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite p > 1. We consider socalled order 2 digital (t, d)-sequences over the finite field with two elements and show that such sequences achieve the optimal order of L p -discrepancy simultaneously for all p ∈ (1, ∞).

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“…In [11] it was shown that the same conditions are also sufficient and necessary for L p (R Σ b,n ) = O( √ log N /N) for all p ∈ [1, ∞). There are also some known facts on the symmetrized point sets as introduced in Definition 1.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 97%

L2 discrepancy of symmetrized generalized Hammersley point sets in base b

Kritzinger

2016

Journal of Number Theory

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Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of L p discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize b-adic Hammersley point sets scrambled with arbitrary permutations. It is already known that these modifications indeed assure that the L p discrepancy is of optimal order O √ log N /N for p ∈ [1, ∞) in contrast to the classical Hammersley point set. We prove an exact formula for the L 2 discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest L 2 discrepancy for every base b ∈ {2, . . . , 27} by employing computer search algorithms.We also introduce the set A b (τ ) = {σ ∈ S b : σ • τ = τ • σ}.

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Lp- and Sp,qr

Cited by 6 publications

References 28 publications

Discrepancy of Second Order Digital Sequences in Function Spaces With Dominating Mixed Smoothness

Discrepancy of Second Order Digital Sequences in Function Spaces With Dominating Mixed Smoothness

Optimal L p -discrepancy bounds for second order digital sequences

L2 discrepancy of symmetrized generalized Hammersley point sets in base b

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