On the best approximation of non-integer constants by polynomials with integer coefficients

Abstract

The exact decrease rate of the best approximations of non-integer numbers by polynomials with integer coefficients of growing degrees is found on a disk in the complex plane, a cube in $\mathbb{R}^{d}$, and a ball in $\mathbb{R}% ^{d}$. The $\sup $-norm is used in the first two cases, and the norm in $% L_{p}$, $1\leq p<\infty $, is applied in the third one. Detailed comments are given (two remarks at the end of the paper).