Irrationality measures for Γ(1/4) and Γ(1/3)
Abstract
Let μ denote the irrationality exponent. Bruiltet proved in 2002 that μ(Γ(1/4))≤ 10143 and μ(Γ(1/3))≤ 10151. We prove μ(Γ(1/4))≤ 3.24· 1026 and μ(Γ(1/3))≤ 1.74· 1026, with explicit thresholds h(p/q)≥ 5.4· 107 and h(p/q)≥ 4.7· 107 respectively, against 1075 and 1079 in the published statements. The proof keeps the architecture of Bruiltet's argument -- Philippon's transcendence criterion together with a zero estimate in commutative algebraic groups -- and replaces three of its ingredients. First, an approximate Schwarz principle based on Hermite interpolation replaces the gravitation estimate of Reyssat; on a lattice point set its cost per node is governed by a lattice energy and drops from about 3.7 to about 1.0. Second, a summation lemma for the failure of Diaz's approximation lemma sharpens the passage from an approximation measure to an irrationality measure. Third, the auxiliary point is taken at a torsion point of order 4, respectively 3, which lowers the degree of the base field from 8d to 2d. All numerical constants are produced by a verifier run in interval arithmetic, supplied as ancillary files.
Related
BibTeX
@misc{giraudin2026irrationality,
title = {{Irrationality measures for $Γ(1/4)$ and $Γ(1/3)$}},
author = {David Niedbala Giraudin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01711},
eprint = {2611.01711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01711},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irrationality measures for Γ(1/4) and Γ(1/3)
AU - David Niedbala Giraudin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01711
UR - https://arxiv.org/abs/2611.01711
AB - Let μ denote the irrationality exponent. Bruiltet proved in 2002 that μ(Γ(1/4))≤ 10^143 and μ(Γ(1/3))≤ 10^151. We prove μ(Γ(1/4))≤ 3.24· 10^26 and μ(Γ(1/3))≤ 1.74· 10^26, with explicit thresholds h(p/q)≥ 5.4· 10^7 and h(p/q)≥ 4.7· 10^7 respectively, against 10^75 and 10^79 in the published statements. The proof keeps the architecture of Bruiltet's argument -- Philippon's transcendence criterion together with a zero estimate in commutative algebraic groups -- and replaces three of its ingredients. First, an approximate Schwarz principle based on Hermite interpolation replaces the gravitation estimate of Reyssat; on a lattice point set its cost per node is governed by a lattice energy and drops from about 3.7 to about 1.0. Second, a summation lemma for the failure of Diaz's approximation lemma sharpens the passage from an approximation measure to an irrationality measure. Third, the auxiliary point is taken at a torsion point of order 4, respectively 3, which lowers the degree of the base field from 8d to 2d. All numerical constants are produced by a verifier run in interval arithmetic, supplied as ancillary files.
ER -