On generalized non-uniform Korobov grids

Generalized non-uniform Korobov grids are considered in the paper.
Three new constructions are considered: the product of non-uniform grids by mutually simple modules; modified non-uniform grids; the product of an uneven grid and a parallelepipedal grid by a mutually simple module.
A paradoxical result is established about the value of the mathematical expectation of the error of approximate integration over modified non-uniform grids.
It is shown that the algorithm of approximate integration using the product of an uneven grid and a parallelepipedal grid in a mutually simple module is unsaturated with the order $\frac{\alpha}{2}$.