Given a regular compact set $E$ in the complex plane, a unit measure $mu$ supported by $partial E,$ a triangular point set $beta := {{beta_{n,k}}_{k=1}^n}_{n=1}^{infty},betasubset partial E$ and a function $f$, holomorphic on $E$, let $pi_{n,m}^{beta
,f}$ be the associated multipoint $beta-$ Pade approximant of order $(n,m)$. We show that if the sequence $pi_{n,m}^{beta,f}, ninLambda, m-$ fixed, converges exact maximally to $f$, as $ntoinfty,ninLambda$ inside the maximal domain of $m-$ meromorphic continuability of $f$ relatively to the measure $mu,$ then the points $beta_{n,k}$ are uniformly distributed on $partial E$ with respect to the measure $mu$ as $ ninLambda$. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence $Lambda$ is provided, under the condition that $Lambda$ is dense enough.
We introduce and analyze some numerical results obtained by the authors experimentally. These experiments are related to the well known problem about the distribution of the zeros of Hermite--Pade polynomials for a collection of three functions $[f_0
equiv 1,f_1,f_2]$. The numerical results refer to two cases: a pair of functions $f_1,f_2$ forms an Angelesco system and a pair of functions $f_1=f,f_2=f^2$ forms a (generalized) Nikishin system. The authors hope that the obtained numerical results will set up a new conjectures about the limiting distribution of the zeros of Hermite--Pade polynomials.
