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) :=

No abstract

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base b. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness S r p,q B([0, 1) s ), which will also give us bounds on the L p -discrepancy. Our sequence and point sets will achieve the known optimal order for the L p -and S r p,q B-discrepancy. The results in this paper generalize several previous results on L p -and S r p,q B-discrepancy estimates and provide a sharp upper bound on the S r p,q B-discrepancy of one-dimensional sequences for r > 0. We will use the b-adic Haar function system in the proofs.

We use the Haar function system in order to study the L 2 discrepancy of a class of digital (0, n, 2)-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We will obtain such formulas not only for the classical digital nets, but also for shifted and symmetrized versions thereof. The basic idea of our proofs is to calculate all Haar coefficents of the discrepancy function exactly and insert them into Parseval's identity. We will also discuss reasons why certain (symmetrized) digital nets fail to achieve the optimal order of L 2 discrepancy and use the Littlewood-Paley inequality in order to obtain results on the L p discrepancy for all p ∈ (1, ∞).

No abstract