Irrationality exponents of logarithms of positive rational numbers

Abstract

We study rational approximation to logarithms of positive rational numbers and give a separated-weight determinant argument for $\mu (\log r)=2$ whenever $r\in {\mathbb{Q}}_{>0}\setminus \{1\}$. The geometric input is an interpolation theorem for logarithmic jets at points with distinct multiplicative coordinates. Its successive weight thresholds are independent of the interpolation centres; the eventual degree threshold may depend on them. A local intersection-length bound and a curve inequality lead to surjectivity through a blow-up and Serre vanishing. For a fixed rational argument, this produces a nonzero rational determinant. Clearing denominators gives an arithmetic lower bound, while an exact row translation and repeated Taylor indices yield a contradictory analytic upper bound. The consequences include nonzero rational linear combinations of such logarithms, rational affine changes, and logarithms of positive numbers with a rational positive integer power. We also give a bounded-fibre interpolation extension, an explicit bound for the final analytic degree, and a counterexample to interpolation under the volume condition alone.

arXiv comment18 pages. Revised manuscript and abstract, with expanded historical context, explanations of the parameter choices, and a discussion of the connection with the argument for pi through the exponential map