About the method of estimating critical determinants within the question of the estimation of the constant of simultaneous diophantine approximations
This paper is devoted to the estimation of the constant of sumultaneous Diophantine approximations for n real numbers. The approach developed by H. Davenport and J. W. S. Cas-sels. H. Davenport discovered the connection between the value of the critical determinant of a star body and the estimation of some forms. In the particular case, this allows calculating the critical determinant of the (n + 1)-dimensional star body of Davenport F„ : |жо| max |ж/” n,s - the volume of a parallelepiped centered at the origin of coordinates located inside the (n + 1)-dimensional star body 1 S П Fn,s : fn,s = 2^ П |ж* + Xs+i\ ^ \xi\ n,s. Earlier, estimates for Vn,s were obtained in the works of J. W. S. Cassels, T. Cusick, S. Krass. This paper is devoted to methods of forming hypotheses about the values of Vn,s based on the results of numerical experiments. The article outlines the approach to obtaining parallelepipeds contained within a star body and possessing the largest volume. This approach combines the use of both numerical and analytical methods.