This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let , be positive integers with . Let be the space of polynomials of degree at most on with integer coefficients such that and are integers for and let be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of from in . We give rather precise quantitative estimations for successive minima of in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in .