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

Abstract

We prove a general effective result concerning approximation of irrational values at rational points a / b of any G-function F with rational Taylor coefficients by fractions of the form $n/(B\cdot b^m)$, where the integer B is fixed. As a corollary, we show that if F is not in $\mathbb Q(z)$, for any $\varepsilon >0$ and any b and m large enough with respect to a, $\varepsilon$ and F, then $|F(a/b)-n/b^m|\ge 1/b^{m(1+\varepsilon )}$ and $F(a/b)\notin \mathbb Q$. This enables us to obtain a new and effective result on repetition of patterns in the b-ary expansion of $F(a/b^s)$ for any $b\ge 2$. In particular, defining $\mathcal {N}(n)$ 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 $\limsup _n \mathcal {N}(n)/n\le \varepsilon$ provided the integer $s\ge 1$ is such that $b^s$ is large enough with respect to a, $\varepsilon > 0$ and F. This improves over the previous bound $1+\varepsilon$, that can be deduced from the work of Zudilin. Our crucial ingredient is the use of non-diagonal simultaneous Padé type approximants for any given family of G-functions solution of a differential system, in a construction à la Chudnovsky-André. This idea was introduced by Beukers in the particular case of the function $(1-z)^\alpha$ in his study of the generalized Ramanujan–Nagell equation, and we use it in its full generality here. In contrast with the classical Diophantine “competition” between E-functions and G-functions, similar results are still not known for a single transcendental value of an E-function at a rational point, not even for the exponential function.