Convergents and irrationality measures of logarithms

Abstract

We prove new irrationality measures with restricted denominators of the form ${\mathrm{d}}_{\lfloor \nu m\rfloor }^s{B}^m$ (where $B,m\in \mathbb{N},\nu >0$, $s\in \{0,1\}$ and ${\mathrm{d}}_m=\text{lcm}\{1,2,\dots ,m\}$) 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 $a,c\ge 1$, for all large enough $b,n$, the denominator $q_n$ of the $n$-th convergent of $\log (1\pm a/b)$ cannot be written under the form ${\mathrm{d}}_{\lfloor \nu m\rfloor }^s(bc{)}^m$: this is true for $a=c=1$, $b\ge 12$ when $s=0$, resp. $b\ge 2$ when $s=1$ and $\nu =1$. Our method rests on a detailed diophantine analysis of the upper Padé table $([p/q]{)}_{p\ge q\ge 0}$ of the function $\log (1-x)$. Finally, we remark that worse results (of this form) are currently provable for the exponential function, despite the fact that the complete Padé table $([p/q]{)}_{p,q\ge 0}$ of $\exp (x)$ and the convergents of $\exp (1/b)$, for $|b|\ge 1$, are well-known, for example.