Metric results on the approximation of zero by linear combinations of independent and of dependent rationals

We give some “rational analoga” to metric results in the classical theory of the diophantine approximation of zero by linear forms. That is: we study the behaviour of expressions of the form $$\begin{gathered} \lim _{m \to \infty } \frac{1}{{\left| {P_s (m)} \right|}}|\{ (x_1 , \ldots ,x_s ) \in P_s (m): \hfill \\ \parallel a_1 \frac{{x_1 }}{m} + \ldots + a_s \frac{{x_s }}{m}\parallel _m \geqslant \psi (a_1 , \ldots ,a_s ,m) \hfill \\ for all - \frac{m}{2}< a_1 , \ldots ,a_s \leqslant \frac{m}{2}, \hfill \\ with (a_1 , \ldots ,a_s ) \ne (0, \ldots ,0)\} |, \hfill \\ \end{gathered} $$ whereP _s (m) is a certain subset of {1, …,m} ^s , ψ is a certain nonnegative function, and ‖ · ‖ _m means the maximum of 1/m and the distance to the nearest integer. Some of the investigations are also motivated by problems in the theory of uniform distribution and of pseudo-random number generation. The results partly depend on the validity of the generalized Riemann hypothesis.