From the introduction: Let \(K\) be a compact subset of the complex plane containing more than one point whose complement \(\Omega =: \overline{\mathbb{C}} \setminus K \) is a simply connected domain. Denote by \(\Psi: \mathbb D^{*} \to \Omega\), the Riemann conformal mapping from \(\mathbb D^{*} : = \{w \in \overline{\mathbb{C}} \,: |w| > 1 \}\) onto \(\Omega\) such that \(\Psi(\infty) = \infty\) and \( \Psi^{\prime}(\infty) > 0\). The \(n\)th Faber polynomial \(F_n = F_{n,K}\), \(n=0,1,2,\dots \), with respect to \(K\) is defined to be an appropriate coefficient of a Laurent series \[ \frac{\Psi^{\prime}(w)}{\Psi(w) - z}= \sum_{n=0}^\infty\frac{F_n(z)}{w^{n+1}}, \quad \quad z \in K. \] The authors study the asymptotic behavior of the distribution of zeros \(Z_n = Z_{n,K} : = \{ z \in \mathbb C \,: F_n(z) = 0 \}\) as \(n \to \infty \). If \(K\) has empty interior, a consequence of a result due to \textit{J. L. Ullman} [Trans. Am. Math. Soc. 94, 515--528 (1960; Zbl 0090.04704)] states that just the points of \(K\) attract zeros of Faber polynomials. Let \(\nu_n\), \(n \in \mathbb N\), denote the normalized counting measure for \(Z_n\), i.e., \(\nu_n = \nu_{n,K} : = \frac{1}{n}\sum_{z \in Z_n} \delta_z \), \(\delta_z\) being the unit Dirac measure concentrated at \(z \in \mathbb C\) and the zeros are counted according to their multiplicities. \textit{A. B. J. Kruijlaars} and \textit{E. B. Saff} [Math. Proc. Camb. Philos. Soc. 118, No. 3, 437--447 (1995; Zbl 0848.30004)] strengthened the Ullman result showing that if \(K\) has empty interior then the sequence \(\{\nu_n\}_{n=1}^{\infty}\) converges to the equilibrium measure \(\mu = \mu_K\) for \(K\) in the weak-star topology. In the paper under review, a modern potential-theoretic interpretation is established which allows the authors to obtain a quantitative extension of the Kuijlaars-Saff result: Let \(K = L\) be a smooth Jordan arc with endpoints \(\zeta_1 \) and \(\zeta_2\), \(\zeta_1 \neq \zeta_2\). For \(\xi \in \Omega \setminus \{\infty\}\), let \[ \xi_L : = \Psi \left( \frac{\Phi(\xi)}{|\Phi(\xi)|}\right)\,, \] where \(\Phi = \Psi^{-1}\), and let \( S(J) : = \{ \xi \in \Omega \setminus{\infty} \,: \xi_L \in J\}\), \(J \subset L\). In this setting, the main result of the paper (Theorem 1) states that if \(L\) is a Dini-smooth arc, then, for any arc \(J \subset L\) and \(n \in \mathbb N\), the inequality \[ \left | \nu_n(\overline{S(J)}) - \mu_L(J) \right | \leq \frac{c}{n} \] holds with a a constant \(c\) depending only on \(L\). A crucial ingredient in the proof of this result is a generalization of a classical theorem of \textit{P. Erdős} and \textit{P. Turán} [Indag. Math. 10, 370--378, 406--413 (1948; Zbl 0031.25402; Zbl 0032.01601)] describing discrepancy estimates between the normalized zero counting measure of a monic polynomial \(p_n\) and the equilibrium distribution of the unit disc if the norm of the polynomial is given on a smaller (resp. larger) circle \(K_{\theta} = \{ z \,: |z| = \theta \}\), \(0 < \theta < 1\) (resp. \(1 < \theta <\infty\)), and all zeros of \(p_n\) lie outside the open unit disk (resp. on the unit disk). This generalization is obtained by using certain functions defined in term of the harmonic measures which are approximated by real parts of complex polynomials.