Abstract
We study the periodic L_2-discrepancy of point sets in the d-dimensional torus. This discrepancy is intimately connected with the root-mean-square L_2-discrepancy of shifted point sets, with the notion of diaphony, and with the worst-case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In discrepancy theory, many results are based on averaging arguments. In order to make such results relevant for applications, one requires explicit constructions of point sets with “average” discrepancy. In our main result, we study Korobov’s p-sets and show that this point sets have periodic L_2-discrepancy of average order. This result is related to an open question of Novak and Woźniakowski.
Highlights
We study the periodic L2-discrepancy which is a quantitative measure for the irregularity of distribution of a point set, but which is closely related to the worst-case integration error of quasi-Monte Carlo integration rules
This means that the term 3/2 in the lower bounds in Theorem 2 is the exact basis for the exponential dependence of the information complexity in d
Summary
We study the periodic L2-discrepancy which is a quantitative measure for the irregularity of distribution of a point set, but which is closely related to the worst-case integration error of quasi-Monte Carlo integration rules (see, for example, [3,4,5,9]). The local discrepancy of a point set P = {x1, x2, . XN } consisting of N elements in the d-dimensional unit cube with respect to a periodic box B = B(x, y) is given by ΔP (B). WN }, the local discrepancy of the weighted point set. The periodic L2-discrepancy of the weighted point set P is. The root-mean-square L2-discrepancy of a shifted (and weighted) point set P with respect to all uniformly distributed shift vectors δ ∈ [0, 1)d is.
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