Good lattice rules based on the general weighted star discrepancy

We study the problem of constructing rank-1 lattice rules which have good bounds on the “weighted star discrepancy”. Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used later to prove that these lattice rules may be obtained using a component-by-component (CBC) construction of the generating vector. Under appropriate conditions on the weights, these lattice rules satisfy strong tractability bounds on the weighted star discrepancy. Particular classes of weights known as “order-dependent” and “finite-order” weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights.