On a functional analog of the Thue-Siegel-Roth theorem

Let $f$ be a germ (the power series expansion) of an algebraic function at infinity. We discuss the limiting properties of the convergents of a functional continued fraction with polynomial coefficients for $f$ (alternative names are diagonal Pade approximants or best local rational approximants). If we compare this functional continued fraction for $f$ with the usual continued fraction (with integer coefficients) for a real number, then the degree of the polynomial coefficient is analogous to the value (magnitude) of the integer coefficient. In our joint work with M. Yattselev [1], we derived strong (or Bernshtein-Szegö type) asymptotics for the denominators of the convergents of a functional continued fraction for analytic function with a finite number of branch points (which are in a generic position in the complex plane). One of the applications following from this result is a sharp estimate for a functional analog of the Thue-Siegel-Roth theorem.