  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 $L_p$, $1\leq p<\infty$. Comments are also given (two
remarks in the end of the paper).
