I'm pretty sure a machine translation of the paper from russian into english will be sufficient. For starters, I fed two paragraphs on pages 417-418 to https://deepl.com, with the following output (unedited, the only changes I made were to LaTeX the symbols):
The top and bottom estimates in Korobov's paper, as well as in the present paper, differed by a constant. Apparently, there are no other nontrivial examples of systems consisting of more than two numbers for linear forms from which the top and bottom estimates would be so close to each other. For linear forms (8), except for the case discussed in [5], the top estimates of only $C_2H^{-s}$ were known. These estimates are derived by means of the Dirichlet principle and do not take into account the specificity of function (1).
The method of proving the theorem will be partially based on the effective construction of the system of linear approximation forms [3]. However, here the system of approximating forms will be $s$ times more "dense," and compared to previous works, the process of exclusion will be changed.
The method applied allows us to effectively find all primitive forms for which (10) holds for $H>H_0(C_2,a,b,\lambda_1,\ldots,\lambda_s)$.
The top and bottom estimates in Korobov's paper, as well as in the present paper, differed by a constant. Apparently, there are no other nontrivial examples of systems consisting of more than two numbers for linear forms from which the top and bottom estimates would be so close to each other. For linear forms (8), except for the case discussed in [5], the top estimates of only $C_2H^{-s}$ were known. These estimates are derived by means of the Dirichlet principle and do not take into account the specificity of function (1).
The method of proving the theorem will be partially based on the effective construction of the system of linear approximation forms [3]. However, here the system of approximating forms will be $s$ times more "dense," and compared to previous works, the process of exclusion will be changed.
The method applied allows us to effectively find all primitive forms for which (10) holds for $H>H_0(C_2,a,b,\lambda_1,\ldots,\lambda_s)$.