Rational approximation to values of G-functions, and their expansions in integer bases

Abstract : Building upon previous works of André and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B ·b^m), where the integer B is fixed. As a corollary, we show that if F is not in Q(z), then for any ε > 0, |F (a/b) − n/b^m | ≥ 1/b^{m(1+ε)} provided b and m are large enough with respect to a, ε and F. This enables us to obtain a new result on the repetition of patterns in the b-ary expansion of F (a/b) when b ≥ 2. In particular, defining N (n) as the number of consecutive equal digits in the b-ary expansion of F (a/b^s) starting from the n-th digit, we prove that lim sup N (n)/n ≤ ε provided the integer s ≥ 1 is such that b s is large enough with respect to a, ε and F. This is a step towards the conjecture that this limit should be equal to 0 whenever F (a/b) is an irrational number. All our results are effective.
Type de document :
Pré-publication, Document de travail
IF_PREPUB. 2017
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https://hal.archives-ouvertes.fr/hal-01245428
Contributeur : Tanguy Rivoal <>
Soumis le : mardi 10 octobre 2017 - 19:27:47
Dernière modification le : jeudi 11 janvier 2018 - 06:12:19
Document(s) archivé(s) le : jeudi 11 janvier 2018 - 14:29:27

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approxGDef.pdf
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  • HAL Id : hal-01245428, version 2
  • ARXIV : 1512.06534

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Stéphane Fischler, Tanguy Rivoal. Rational approximation to values of G-functions, and their expansions in integer bases. IF_PREPUB. 2017. 〈hal-01245428v2〉

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