Proof techniques in quasi-Monte Carlo theory

Dick, Josef; Hinrichs, Aicke; Pillichshammer, Friedrich

doi:10.1016/j.jco.2014.09.003

Cited by 21 publications

(10 citation statements)

References 92 publications

(158 reference statements)

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“…We assume some familiarity with the proof of Roth in the language of Haar functions, as can be found, e.g., in [2] or [6]. We only prove the case d = 2; the extension to general d is done as for Roth…”

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confidence: 99%

Extreme and periodic $L_2$ discrepancy of plane point sets

Hinrichs

Kritzinger²,

Pillichshammer³

2021

Acta Arith.

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We study several discrepancy notions of two wellknown instances of plane point sets, namely the Hammersley point set and rational lattices. The discrepancies are considered with respect to the L 2 norm and a variety of test sets. We define (standard) L 2 discrepancy, extreme L 2 discrepancy and periodic L 2 discrepancy. Let P = {x 0 , x 1 ,. .. , x N −1 } be an arbitrary N-element point set in the unit square [0, 1) 2. For any measurable subset B of [0, 1] 2 we define the counting function A(B, P) := n ∈ {0, 1,. .. , N − 1} : x n ∈ B , i.e., the number of elements from P that belong to the set B. By the local discrepancy of P with respect to a given measurable "test set" B one understands the expression A(B, P) − N λ(B), where λ denotes the Lebesgue measure of B. A global discrepancy measure is then obtained by considering a norm of the local discrepancy with respect to a fixed class of test sets. Here we restrict ourselves to the L 2 norm, but we vary the class of test sets. The (standard) L 2 discrepancy uses the class of axis-parallel squares anchored at the origin as test sets. The formal definition is L 2,N (P) :=

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confidence: 99%

Extreme and periodic $L_2$ discrepancy of plane point sets

Hinrichs

Kritzinger²,

Pillichshammer³

2021

Acta Arith.

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“…Proof. We assume some familiarity with the proof of Roth in the language of Haar functions as it can be found, e.g., in [2] or [6]. We only prove the case d = 2, the extension to general d is done as for Roth's lower bound.…”

Section: Proposition 3 Formentioning

confidence: 99%

Extreme and periodic $L_2$ discrepancy of plane point sets

Hinrichs¹,

Kritzinger²,

Pillichshammer³

2020

Preprint

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In this paper we study the extreme and the periodic L 2 discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic L 2 discrepancy is, up to a multiplicative factor, also known as diaphony. The main results are exact formulas for these kinds of discrepancies for the Hammersley point set and for rational lattices.In order to value the obtained results we also prove a general lower bound on the extreme L 2 discrepancy for arbitrary point sets in dimension d, which is of order of magnitude (log N ) (d−1)/2 , like the standard and periodic L 2 discrepancies, respectively. Our results confirm that the extreme and periodic L 2 discrepancies of the Hammersley point set are of best possible asymptotic order of magnitude. This is in contrast to the standard L 2 discrepancy of the Hammersley point set. Furthermore our exact formulas show that also the L 2 discrepancies of the Fibonacci lattice are of the optimal order. We also prove that the extreme L 2 discrepancy is always dominated by the standard L 2 discrepancy, a result that was already conjectured by Morokoff and Caflisch when they introduced the notion of extreme L 2 discrepancy in the year 1994.

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“…With standard techniques one can prove a Koksma-Hlawka inequality according to D π (w, P n ). For details we refer to [7], […”

Section: Integration Error and Weighted Star-discrepancymentioning

confidence: 99%