Approximations of algebraic functions by rational ones — functional analogues of diophantine approximants

A goal of this note is to discuss applications of our result on asymptotics of the convergents of a continued fraction of an analytic function with branch points. We consider famous problems: on normality of the Pade approximants for algebraic functions (a functional analog of the Thue–Siegel–Roth theorem and $\varepsilon = 0$ Gonchar–Chudnovskies conjecture), on estimation of the number of “spurious” (“wandering”) poles of rational approximants for algebraic functions (Stahl conjecture), on appearance and disappearance of the Froissart doublets.