Let \(\mathbb A_n\) be the set of real algebraic numbers of degree \(n\) and \(H(\alpha)\) be the height of \(\alpha\in \mathbb A_n\). A ‘best possible’ regular system \((\mathbb A_n,N)\), where \(N(\alpha)= (H(\alpha)/ (1+|\alpha|)^n)^{n+1}\), is obtained for approximating real numbers by numbers \(\alpha\) in \(A_n\). In an interesting application, the author uses this regular system to obtain the following analogue of Groshev’s theorem. Let \(\Psi (q)\), \(q=1,2,\dots\), be a decreasing sequence of positive real numbers such that \[ \sum_{q=1}^\infty \Psi(q) \tag \(*\) \] diverges. Then for almost all real \(x\), the inequality \(|x-\alpha|< H(\alpha)^{-n+1} \Psi(H(\alpha))\) holds for infinitely many \(\alpha\in \mathbb A_n\). This is in turn exploited to prove the corresponding result for polynomials: For almost all real \(x\), the inequality \[ |P(x)|< H(P)^{-n+1} \Psi(H(P)), \tag \(**\) \] where \(H(P)\) is the height of \(P\), holds for infinitely many integral polynomials \(P\) of degree at most \(n\). The converse that when the sum \((*)\) converges, the inequality \((**)\) has only finitely many solutions for almost all \(x\) was conjectured by A. Baker and proved by V. I. Bernik [Acta Arith. 53, 17–28 (1989; Zbl 0692.10042); see also Chapter 2 in V. I. Bernik and M. M. Dodson, Metric number theory, Cambridge University Press (1999; Zbl 0933.10040)]. A similar result has been established for \(C^3\) planar curves with non-zero curvature almost everywhere [V. V. Beresnevich, V. I. Bernik, H. Dickinson and M. M. Dodson, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1988, 3053–3063 (1999; Zbl 0978.11034)].