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Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 29, 2016 17:30, Dorodnitsyn Computing Centre, Department of Mechanics and Mathematics of Lomonosov Moscow State University., 119991, Moscow, Gubkina str., 8, Steklov Mathematical Institute, 9 floor, Conference hall

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

A. I. Aptekarev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
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A. I. Aptekarev

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Abstract: 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.

Language: Russian and English

  1. A.I. Aptekarev, M.L. Yattselev, “Padé approximants for functions with branch points — strong asymptotics of Nuttall-Stahl polynomials”, Acta Math., 215:2 (2015), 217–280, arXiv: 1109.0332v2 [math.CA]  crossref  mathscinet  isi

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