This paper presents a historical review of the results on the problem of estimating a constant
best joint diophantine approximations for $ n $ real numbers. This problem
is a special case of the more general problem of approximating $ n $ real linear forms and has itself rich history,
ascending to P. G. Dirichlet. We will focus more on the approach of H. Davenport, based
on the connection of Diophantine approximations with the geometry of numbers.The first part provides an overview of
results obtained for $ n = 1 $ and $ n = 2 $ real numbers.
Historically, estimates for $ n = 1 $ are based on the theory of continued fractions and the most significant is
A. Hurwitz's estimate obtained in 1891. For $ n = 2 $ the basis of known estimates is
mathematical apparatus of linear algebra (F. Furtwangler),
geometry of numbers (H. Davenport, J. W. S. Cassels) and the results of multidimensional generalizations of continued fractions
(V. Adams, T. Cusick).The second part is devoted to one of the first general estimates from below obtained in 1929 by F. Furtwangler.
He constructed estimates of the discriminants of algebraic fields, which lead to an estimate of the quality of the approximation
of $ n $ real numbers by rational, which in turn results in estimating the constant
best joint diophantine approximations.The third part outlines the most fundamental of the currently known assessments obtained
H. Davenport and then refined by J. W. S. Cassels. H. Davenport discovered the connection between the value
critical determinant of lattices and the estimating of some forms. In the particular case it allows
to get the value of the constant best joint diophantine approximations by
calculating the critical determinant of the special lattice. However the calculation of critical determinants
for lattices of this type is a difficult task. Therefore, J. W. S. Cassells switched
from the direct calculation of the critical determinant to the estimating of its value.
This approach proved to be quite fruitful and allowing us to obtain estimates of the constant
best joint diophantine approximations for $ n = 2, 3, 4 $.The fourth part gives an overview of the well-known estimates from below for $ n > 2 $.
These results are based on the use of the aforementioned approach of J. W. S. Cassels.
It is worth noting that assessments of this kind are quite a complicated computational problem
and in each case the solution of this problem required the use of new approaches.In the last part we present a review of some well-known estimates of the constant
best diophantine approximations from above. Although this problem is not the main topic of this
articles, but considerable interest is the comparison of approaches in assessing
constants of the best joint diophantine approximations from above and below.
The first estimate from above was obtained by H. Minkowski in 1896 using the geometry of numbers.
H. F. Blichfeldt introducing the concept of the fundamental parallelepiped in 1914
improved the result of H. Minkowski. Later the approach of H. F. Blichfeldt received
development in the works of P. Mullender, W. G. Spohn, W. G. Nowak.
