Discrepancy Bounds for Mixed Sequences

A mixed sequence is a sequence in the $s$-dimensional unit cube which one obtains by concatenating a $d$-dimensional low-discrepancy sequence with an $s-d$-dimensional random sequence. We discuss some probabilistic bounds on the star discrepancy of mixed sequences

Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces

We extend the notion of L2-B-discrepancy introduced in [E. Novak, H. Wozniakowski, L2 discrepancy and multivariate integration, in: Analytic number theory. Essays in honour of Klaus Roth. W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, and R. C. Vaughan (Eds.), Cambridge University Press, Cambridge, 2009, 359"“388] to what we want to call weighted geometric L2-discrepancy. This extended notion allows us to consider weights to moderate the importance of different groups of variables, and additionally volume measures different from the Lebesgue measure as well as classes of test sets different from measurable subsets of Euclidean spaces. We relate the weighted geometric L2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wozniakowski. Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by algorithms

Discrepancy of Symmetric Products of Hypergraphs

For a hypergraph ${\mathcal H} = (V,{\mathcal E})$, its $d$--fold symmetric product is $\Delta^d {\mathcal H} = (V^d,\{E^d |E \in {\mathcal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${disc}(\Delta^d {\mathcal H},2) \le {disc}({\mathcal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${disc}(\Delta^d {\mathcal H},c) = \Omega_d({disc}({\mathcal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\mathcal H}^d$, which satisfies ${disc}({\mathcal H}^d,c) = O_{c,d}({disc}({\mathcal H},c)^d)$.Comment: 12 pages, no figure