Irrationality measures for $Γ(1/4)$ and $Γ(1/3)$

Abstract

Let $μ$ denote the irrationality exponent. Bruiltet proved in 2002 that $μ(Γ(1/4))\le 10^{143}$ and $μ(Γ(1/3))\le 10^{151}$. We prove $μ(Γ(1/4))\le 3.24\cdot 10^{26}$ and $μ(Γ(1/3))\le 1.74\cdot 10^{26}$, with explicit thresholds $h(p/q)\ge 5.4\cdot 10^{7}$ and $h(p/q)\ge 4.7\cdot 10^{7}$ 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.