On the best approximation of constants by polynomials with integer coefficients

In this paper, exact rate of decrease of best approximations of non-integer numbers by polynomials with integer coefficients of the growing exponentials is found on a disk in complex plane, on a cube in ${\mathbb{R}}^d$, and on a ball in ${\mathbb{R}}^d$. While in the first two cases the $\sup$-norm is used, the third one is fulfilled in $L_p$, $1\le p<\mathrm{\infty }$. Comments are also given (two remarks in the end of the paper).