Rational approximation of the algebraic functions and functional analogues of the Diophantine approximations

Let f be a germ (the power series expansion) of an algebraic function at infinity. We discuss the limiting properties of the convergent of a functional continued fraction with polynomial coefficients for f (alternative name is diagonal Pade approximant or best local rational approximant). 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 convergent of the 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. The bounds on the incomplete quotients for the functional continued fractions of the algebraic functions follows from thus result as well.

References

1. A. I. Aptekarev, M. L. Yattselev, “Pade approximants for functions with branch points – strong asymptotics of Nuttall-Stahl polynomials”, Acta Math., 215:2 (2015), 217–280, arXiv: 1109.0332v2