Convergents and Irrationality Measures of Logarithms

We prove new irrationality measures with restricted denominators of the form dbνmcB m (where B,m ∈ N, ν > 0, s ∈ {0, 1} and dm = lcm{1, 2, . . . , m}) for values of the logarithm at certain rational numbers r > 0. In particular, we show that such an irrationality measure of log(r) is arbitrarily close to 1 provided r is sufficiently close to 1. This implies certain results on the number of non-zero digits in the b–ary expansion of log(r) and on the structure of the denominators of convergents of log(r). No simple method for calculating the latter is known. For example, we show that, given integers a, c ≥ 1, for all large enough b, n, the denominator qn of the n–th convergent of log(1±a/b) cannot be written under the form dbνmc(bc) : this is true for a = c = 1, b ≥ 12 when s = 0, resp. b ≥ 2 when s = 1 and ν = 1. Our method rests on a detailed diophantine analysis of the upper Padé table ([p/q])p≥q≥0 of the function log(1 − x). Finally, we remark that worse results (of these form) are currently provable for the exponential function, despite the fact that the complete Padé table ([p/q])p,q≥0 of exp(x) and the convergents of exp(1/b), for |b| ≥ 1, are well-known, for example.