The author studies the distribution in \([0,1[^ d\) of sequences of the type \((1)\quad (a_{1,j}x_ 1,...,a_{d,j}x_ d)_{j\geq 1} mod 1\) with \((a_{i,j})_{j\geq 1}\) a strictly increasing sequence of integers and \(x_ i\in {\mathbb{R}}\), \(1\leq i\leq d\), and, more general, \((2)\quad ((g_{i,j}(x_ 1),...,g_{d,j}(x_ d))_{j\geq 1}mod 1\) with \(g_{i,j}\) real valued and differentiable. For sequences of the type (1) he generalizes a metric result of R. C. Baker [J. Lond. Math. Soc., II. Ser. 24, 34-40 (1981; Zbl 0422.10048)]. In the proof, a maximal inequality for partial sums of Fourier series in several variables established by P. Sjölin [Ark. Math. 9, 65-90 (1971; Zbl 0212.417)] is applied to show:
Theorem 1. \(D(N,x)=o(N^{-1/2}(\log N)^{1/2+d+\epsilon})\) a.e., where D(N,x) denotes the (usual) discrepancy of the first N points of the sequence (1) in \([0,1[^ d\) and \(x=(x_ 1,...,x_ d)\). Further, it is proved Theorem 2. If \(B=\cup^{\infty}_{k=1}R_ k\), \((R_ k)_{k\geq 1}\) a sequence of disjoint boxes \(R_ k\) in \([0,1[^ d\) where the volume of \(R_ k\) decreases exponentially in k, then for the discrepancy function the estimate \[ | (1/N)\sum_{j\leq N}1_ B((a_{1,j}X_ 1,...,a_{d,j}x_ d))-volume(B)| \quad <\quad N^{1/2}(\log N)^{d+3/2+\epsilon} \] holds. Again, this generalizes (and, here, even improves) a result of R. C.Baker [Mathematika 21, 248-260 (1975; Zbl 0296.10037)]. The paper is completed by further metric results on the distribution of the sequence of type (2), for example an estimate of the Hausdorff dimension of the set \(\{\) \(x\in X :\limsup_{N\to \infty} N^ q D(N,x)>0\}\), \(0<q<1/2\).