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

Abstract

Building upon previous works of Andr{\'e} 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 $\times$b^m), where the integer B is fixed. As a corollary, we show that if F is not in Q(z), then for any $\epsilon$ > 0, |F (a/b) -- n/b^m | $\ge$ 1/b^{m(1+$\epsilon$)} provided b and m are large enough with respect to a, $\epsilon$ 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 $\ge$ 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 $\le$ $\epsilon$ provided the integer s $\ge$ 1 is such that b s is large enough with respect to a, $\epsilon$ 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.