This talks consists of two parts. In the first part, we discuss recent results on the relation between discrepancy theory, metric number theory, and the almost everywhere convergence of series of dilated functions. In particular, we describe the almost independent behavior of lacunary function systems, and present new upper bounds for certain sums involving greatest common divisors.
In the second part, we deal with the tractability of Quasi-Monte Carlo integration in a high-dimensional setting. We present upper and lower bounds for the inverse of the star-discrepancy, reconsider a conjecture of Novak and Wozniakowski, and discuss possible methods for further improvements of the known bounds.
Christoph Aistleitner is currently an Erwin Schrödinger fellow of the Austrian Science Fund FWF. He will be working as a guest researcher in the Department of Applied Mathematics, UNSW, for one year.