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.
Document type :
Preprints, Working Papers, ...
Liste complète des métadonnées

Cited literature [20 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01245428
Contributor : Tanguy Rivoal <>
Submitted on : Tuesday, October 10, 2017 - 7:27:47 PM
Last modification on : Thursday, January 11, 2018 - 6:12:19 AM
Document(s) archivé(s) le : Thursday, January 11, 2018 - 2:29:27 PM

Files

approxGDef.pdf
Files produced by the author(s)

Identifiers

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

Citation

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

Share

Metrics

Record views

114

Files downloads

44