Digital nets in dimension two with the optimal order of L_p discrepancy

Abstract

We study the L p 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 L 2 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 L p discrepancy for all p ∈ [1, ∞).Our findings are in the spirit of a paper by Bilyk from 2013, who considered the L 2 discrepancy of lattices consisting of the elements (k/N, {kα}) for k = 0, 1, . . . , N − 1, and who gave Diophantine properties of α which guarantee the optimal order of L 2 discrepancy. Proposition 7 Let j ∈ J 7 and m ∈ D j . Then we have |µ j,m | 2 −n−j 1 −j 2 for all m ∈ D j and |µ j,m | = 2 −2j 1 −2j 2 −4 for all but at most 2 n elements m ∈ D j .Proof. At most 2 n of the 2 |j| dyadic boxes I j,m for m ∈ D j contain points. For the empty boxes, only the linear part of the discrepancy function contributes to the corresponding Haar coefficients; hence |µ j,m | = 2 −2j 1 −2j 2 −4 for all but at most 2 n elements m ∈ D j . The non-empty boxes contain at most 4 points. Hence we find by (11) |µ j,m | ≤2 −n−j 1 −j 2 −2 z∈P∩I j,m

On Quasi-Monte Carlo Methods in Weighted ANOVA Spaces

“…• u = {1, 2}: Here, again according to Lemma 16, Eq. ( 22) is n j=1 Several constructions of two-dimensional projection regular point sets with best possible order of L p * -discrepancy for all p * ∈ [1, ∞] are known, e.g., generalized Hammersley point sets [8], shifted Hammersley point sets [15,26] or digital NUT nets [23]. As an example we would like to present the digitally shifted Hammersley point sets from [15] Furthermore, according to [7,19],…”

Point sets with optimal order of extreme and periodic discrepancy

“…The claim on the extreme L p discrepancy follows from the mentioned results on the relevant Haar coefficients and the second part of Proposition 3. The relevant Haar coefficients of the local discrepancy of the nets P c can be found in [20,Lemma 3.2].…”

“…, 0) in the present Theorem 9). However, the Hammersley point set has a much higher star L 2 discrepancy, which is of order log N, whereas the L 2 discrepancy of P 1 is of optimal order √ log N (see [20]). The large L 2 discrepancy of the Hammersley point set is caused by the Haar coefficient µ (−1,−1),(0,0) .…”