Tractability properties of the weighted star discrepancy of regular grids

In this paper we study tractability properties of the weighted star discrepancy with general coefficients of centered regular grids with different mesh-sizes. We give exact characterizations of the weight sequences $(\gamma_j)_{j \ge 1}$ such that the regular grid with different mesh-sizes achieves weak, uniform weak, quasi polynomial, polynomial or strong polynomial tractability for the $\boldsymbol{\gamma}$-weighted star discrepancy. For example, a necessary and sufficient condition such that the regular grid with different mesh-sizes achieves weak tractability for the $\boldsymbol{\gamma}$-weighted star discrepancy is $\lim_{j \rightarrow \infty}j \gamma_j=0$.