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 μ(log r)=2 whenever r∈ℚ>0∖{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.
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BibTeX
@misc{liu2026irrationality,
title = {{Irrationality exponents of logarithms of positive rational numbers}},
author = {Jingwen Liu and Kai Jiang and Pingwen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10192},
eprint = {2610.10192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10192},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irrationality exponents of logarithms of positive rational numbers
AU - Jingwen Liu
AU - Kai Jiang
AU - Pingwen Zhang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.10192
UR - https://arxiv.org/abs/2610.10192
AB - We study rational approximation to logarithms of positive rational numbers and give a separated-weight determinant argument for μ(log r)=2 whenever r∈ℚ_>0∖{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.
ER -