Irrationality exponents of logarithmic numbers: Values, quotients, and graded approximation
Abstract
We establish degree-dependent irrationality-exponent bounds for logarithms at algebraic arguments and sharper bounds when conjugate logarithms match. A two-torus construction gives an upper bound of three for ratios of logarithms of multiplicatively independent positive rational numbers. We also treat normalized quadratic logarithms and circular logarithms. The rational-logarithm and rational-power exponent-two cases were treated earlier by Liu, Jiang and Zhang; we retain them within the broader framework and its Lean verification. Starting from the interpolation and determinant method in OpenAI's September 24, 2026 preprint on the irrationality exponent of π, we track the arithmetic cost of algebraic embeddings and the effects of matching conjugate logarithms. A second, graded Pade construction gives near-optimal measures for powers and bounded-degree polynomial values at sufficiently small logarithmic arguments, with estimates uniform in both coefficient height and rational denominator. We state the branch, independence, parameter-order, and approximation-field hypotheses explicitly. The Lean supplement has reviewed endpoint correspondences for all 62 numbered results: 53 direct checked endpoints or compositions, and nine applications with named classical theorem inputs. Native proofs of those inputs and separate formalizations of every unnumbered assertion are not supplied. Appendix A gives the exact declarations, hypotheses, and verification evidence.
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BibTeX
@misc{chudnovsky2026irrationality,
title = {{Irrationality exponents of logarithmic numbers: Values, quotients, and graded approximation}},
author = {Gregory V. Chudnovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28079},
eprint = {2610.28079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28079},
}
RIS
TY - UNPB
TI - Irrationality exponents of logarithmic numbers: Values, quotients, and graded approximation
AU - Gregory V. Chudnovsky
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28079
UR - https://arxiv.org/abs/2610.28079
AB - We establish degree-dependent irrationality-exponent bounds for logarithms at algebraic arguments and sharper bounds when conjugate logarithms match. A two-torus construction gives an upper bound of three for ratios of logarithms of multiplicatively independent positive rational numbers. We also treat normalized quadratic logarithms and circular logarithms. The rational-logarithm and rational-power exponent-two cases were treated earlier by Liu, Jiang and Zhang; we retain them within the broader framework and its Lean verification. Starting from the interpolation and determinant method in OpenAI's September 24, 2026 preprint on the irrationality exponent of π, we track the arithmetic cost of algebraic embeddings and the effects of matching conjugate logarithms. A second, graded Pade construction gives near-optimal measures for powers and bounded-degree polynomial values at sufficiently small logarithmic arguments, with estimates uniform in both coefficient height and rational denominator. We state the branch, independence, parameter-order, and approximation-field hypotheses explicitly. The Lean supplement has reviewed endpoint correspondences for all 62 numbered results: 53 direct checked endpoints or compositions, and nine applications with named classical theorem inputs. Native proofs of those inputs and separate formalizations of every unnumbered assertion are not supplied. Appendix A gives the exact declarations, hypotheses, and verification evidence.
ER -