states conditions of convergence on the distribution of nodes.

Let us introduce some previous notation. Set $\mathcal{K}_1$ and $\mathcal{K}_2$ two compact subsets of the complex plane $\mathbb{C}$. Let $\displaystyle \mbox{dist}(\mathcal{K}_1, \mathcal{K}_2)=\min \left\{\left|\left|x-y\right|\right\|: x \in \mathcal{K}_1\,\, \mbox{and}\,\, y\in \mathcal{K}_2\right\}$ denote the distance between $\mathcal{K}_1$ and $\mathcal{K}_2$. Consider a compact set $K \subset \mathbb{C} \setminus [-1,\,1]$, and a measure $\mu$ supported on $K$. A measure $\widetilde{\mu}$ supported on $[-1,\,1]$ is said to be the balayage of $\mu$ if they have the same total variation $||\mu||=||\widetilde{\mu}||$ and their logarithmic potentials coincide on $[-1,\,1]$. This is $$V^{\widetilde{\mu}}(x)=\int \log \frac{1}{|x-t|} \mathrm{d} \widetilde{\mu}(t)=\int \log \frac{1}{|x-\zeta|} \mbox{d} \mu (\zeta)=V^{\mu}(x), \quad x\in [-1,\,1].$$ In [@ST Section II.4] we can find a deep study about balayage of measures. We are now ready to state the main result of this paper:

\[main\] Fix a number $\kappa \in \mathbb{N}$ and a probability discrete measure $$\sigma=\frac{1}{\kappa}\sum_{k=1}^{\kappa} \delta_{\zeta_k}, \quad \zeta_k \subset \mathbb{C}\setminus [-1,\,1], \quad k=1,\ldots, \kappa.$$ Assume that $\sigma$ is symmetric with respect to $\mathbb{R}$ with $\displaystyle \mathrm{dist}\left(\left\{\zeta_1,\ldots,\zeta_{\kappa}\right\},[-1,\,1]\right)>1$. Denote $\widetilde{\sigma}$ the balayage measure associated to $\sigma$ supported on the interval $[-1,\,1]$. Given a rational number $a \in [0,\, 1]$, consider a subsequence $\Lambda \subset \mathbb{N}$ such that for each $n\in \Lambda$, $\displaystyle 2 