Best approximation of constants by polynomials with integer coefficients

What is best approximation of a non-integer number $\lambda \in R$ by polynomials $q_n$ of degree at most $n$ with integer coefficients on the segment $[a,b]\subset (0,1)$ in the uniform metric? In the paper, this old problem is solved for any rational number $\lambda =\frac{p}{q}$ and a segment on the real line. The same problem is solved when the segment is replaced by a disk in $ℂ$ and a cube in $R^m$, both non containing integer points. Best approximation of rational numbers by polynomials with natural coefficients is considered as well. At the same time, the question of uniqueness and non-uniqueness of best approximation polynomials has also been studied. In addition, connection between theorems on best approximation to functions by polynomials with integer coefficients and integer transfinite diameter is established.