On estimating the constant of simultaneous Diophantine approximation

The paper is devoted to the problem of estimating the constant of the best Diophantine approximations. The estimates of lower bound $C_n$ for $n = 5$ and $n = 6$ was improved. The first chapter gives an overview of the history of estimates of the constant of the best Diophantine approximations. In the second chapter sets out the methods and results of numerical experiments leading to estimates of $C_n$. The third chapter is devoted to estimating some functions by means of which in the fourth chapter we will proof the estimates of $C_n$ for $3 \leq n \leq 6$.