Abstract
We study the extreme L_p discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star L_p discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any p>1, the extreme L_p discrepancy of every infinite sequence in [0,1)^d is at least of order of magnitude (log N)^{d/2}, where N is the number of considered initial terms of the sequence. For p in (1,infty ), this order of magnitude is best possible.
Highlights
Let P = {x0, x1, . . . , xN−1} be an arbitrary N -element point set in the d-dimensional unit cube [0, 1)d
The local discrepancy of P with respect to a given measurable “test set” B is given by ΔN (B, P) := AN (B, P) − N λ(B), where λ denotes the Lebesgue measure of B
Using a method from Proınov [25], the results about lower bounds on star Lp discrepancy for finite sequences can be transferred to the following lower bounds for infinite sequences: for every p ∈ (1, ∞] and every d ∈ N, there exists a Cp,d > 0 such that for every infinite sequence Sd in [0, 1)d, Lspt,aNr(Sd) ≥ Cp,d(log N )d/2 for infinitely many N ∈ N
Summary
For an infinite sequence Sd in [0, 1)d, the star and the extreme Lp discrepancies Lp,N (Sd) are defined as Lp,N (Pd,N ), N ∈ N, of the point set Pd,N consisting of the initial N terms of Sd, where Halasz [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 c1,2 with the following property: for every N -element P in [0, 1)2 with N ≥ 2, we have
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