Probability estimates related to Korobov's number-theoretical quadrature formulas

 Let $a_1, \ldots, a_s$ be integers and $N$  be a positive integer. Korobov (1959) and Hlawka (1962) proposed to use the points
$$     x^{(k)} = (\{a_1 k/N\}, \ldots, \{a_1 k/N\}), k=1,\ldots, N, $$
as nodes of multidimensional quadrature formulae. We obtain some new  probability estimates related to  discrepancy of the sequence $K_N(a)=\{x^{(1)},\ldots, x^{(N)}\}$ and   error of Korobov's number-theoretical quadrature formulas.