For \(P_{N,s}=\{\mathbf{x}_0,\ldots, \mathbf{x}_{N-1}\}\subset [0,1)^s\), \(q\in [1,\infty]\), the \(L_q\) discrepancy is \[ L_{q,N}(P_{N,s})=\left(\int_{[0,1]^s} \left|\frac{A_{N}([\mathbf{0}, \mathbf{t}),P_{N,s})}{N}-t_1t_2\cdots t_s\right|^q d\mathbf{t} \right)^{1/q}, \] where \(A_{N}([\mathbf{0}, \mathbf{t}),P_{N,s})\) is the number of \(0\leq n\leq N-1\) with \(\mathbf{x}_n\in [\mathbf{0}, \mathbf{t})=[0,t_1)\times\cdots \times [0, t_s)\). The paper under review is a survey about studies of \(L_q\) discrepancy with \(q\in(1,\infty)\) that begin with K. F. Roth’s general lower bound on the \(L_2\) discrepancy of finite point sets [Mathematika 1, 73–79 (1954; Zbl 0057.28604)]. In particular, the authors give an overview of explicit constructions of point sets and sequences with optimal order of \(L_q\) discrepancy.
For the entire collection see [Zbl 1291.11003].