Extreme and periodic $L_2$ discrepancy of plane point sets

Abstract

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) :=

Exact order of extreme $$L_p$$ discrepancy of infinite sequences in arbitrary dimension

“…Recently it has been shown that the extreme discrepancy is dominated – up to a multiplicative factor that depends only on p and d – by the star discrepancy (see [ 21 , Corollary 5]), i.e., for every and , there exists a positive quantity such that for every and every N -element point set in , we have For , we even have for all ; see [ 17 , Theorem 5]. A corresponding estimate the other way round is in general not possible (see [ 17 , Section 3]). So, in general, the star and the extreme discrepancy for are not equivalent, which is in contrast to the -case.…”

“…For every and , there exists a such that for every finite N -element point set in with , we have For the star discrepancy, the result for is a celebrated result by Roth [ 29 ] from 1954 that was extended later by Schmidt [ 31 ] to the case . For the extreme discrepancy, the result for was first given in [ 17 , Theorem 6] and for , in [ 21 ]. Halász [ 15 ] for the star discrepancy and the authors [ 21 ] for the extreme discrepancy proved that the lower bound is even true for and , i.e., there exists a positive number with the following property: for every N -element in with , we have On the other hand, it is known that for every and every , there exist N -element point sets in such that (For , we write if there exists a positive number C such that for all .…”

“…For p = 2, we even have c d,2 = 1 for all d ∈ N; see [17,Theorem 5]. A corresponding estimate the other way round is in general not possible (see [17,Section 3]).…”

“…For p = 2, we even have c d,2 = 1 for all d ∈ N; see [17,Theorem 5]. A corresponding estimate the other way round is in general not possible (see [17,Section 3]). So, in general, the star and the extreme L p discrepancy for p < ∞ are not equivalent, which is in contrast to the L ∞ -case.…”

“…For the star L p discrepancy, the result for p ≥ 2 is a celebrated result by Roth [29] from 1954 that was extended later by Schmidt [31] to the case p ∈ (1, 2). For the extreme L p discrepancy, the result for p ≥ 2 was first given in [17,Theorem 6] and for p ∈ (1, 2), in [21]. Halász [15] for the star discrepancy and the authors [21] for the extreme discrepancy proved that the lower bound is even true for p = 1 and d = 2, i.e., there exists a positive number c 1,2 with the following property: for every N -element P in [0, 1) 2 with N ≥ 2, we have…”