Convergents and irrationality measures of logarithms (Q925114)

Summary: 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 \(\mathrm{d}_m=\text{lcm}\{1,2, \ldots, m\}\)) 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 \(a,c\geq 1\), 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 \(a=c=1\), \(b \geq 12\) when \(s=0\), resp. \(b \geq 2\) when \(s=1\) and \(\nu=1\). Our method rests on a detailed diophantine analysis of the upper Padé table \(([p/q])_{p\geq q\geq 0}\) 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 \(([p/q])_{p, q\geq 0}\) of \(\exp(x)\) and the convergents of \(\exp(1/b)\), for \(| b| \geq 1\), are well-known, for example.