A compact set \(E\subset{\mathbb C}\) is called uniformly perfect if there exists a constant \(00\) let denote \[ \Omega:=\overline{{\mathbb C}}\setminus E, \] \[ E_\delta:=\{z\in{\mathbb C}:\, g_\Omega(z)=\delta\}, \] \[ \rho_\delta(x):= \text{dist}(x,E_\delta)= \inf_{z\in E_\delta}| z-x| \] and define a function \(r(x,\delta)\) by the relation \( \rho_{r(x, \delta)}(x)=\delta. \) \textbf{Theorem~2.} Let \(E\subset{\mathbb R}\) be uniformly perfect. Suppose that \[ E_n(f,E)=O(n^{-\alpha})\quad\text{as }n\to\infty \] holds for a function \(f\in C(E)\) with some \(00\) is a constant independent of \(x_1\) and \(x_2\). In addition, an extended bibliography will help the reader to find necessary related information. The article is recommended for all researchers in complex analysis and approximation theory.