The irrationality measure of $\Gamma(1/4)$

Description of constant

For a real number $\gamma$, its irrationality exponent $\mu(\gamma)$ is defined by $\mu(\gamma) := \inf\Bigl\{c\in\mathbb{R}:\ \Bigl\lvert\gamma-\frac{a}{b}\Bigr\rvert\le \lvert b\rvert^{-c}\ \text{has only finitely many solutions }(a,b)\in\mathbb{Z}^2\Bigr\}.$

[Zud2004-def-mu]

We define

\[C_{7b}\ :=\ \mu\bigl(\Gamma(1/4)\bigr).\]

For $p/q\in\mathbb{Q}$ in lowest terms with $q>0$, write

\[h(p/q):=\log\max(\lvert p\rvert,q).\]

This agrees with the absolute logarithmic height used by Bruiltet. [Suk2013-def-hpq] [Bru2002-def-h] [Bru2002-def-places]

Known upper bounds

Bound	Reference	Comments
$10^{143}$	[Bru2002]	Bruiltet proves an explicit inequality of the form $h(p/q)\ge 10^{75}\Rightarrow \lvert \Gamma(1/4)-p/q\rvert > (1/(qe))^{10^{143}}$, which implies $\mu(\Gamma(1/4))\le 10^{143}$. [Bru2002-cor-gamma14]

Known lower bounds

Bound	Reference	Comments
$2$	Trivial (Dirichlet)	Every irrational number has irrationality exponent at least $2$.

Additional comments and links

The large gap between the proven upper bound $10^{143}$ and the universal lower bound $2$ reflects the current weakness of methods for proving sharp irrationality measures for special constants such as $\Gamma(1/4)$.
Wikipedia page on irrationality measure

References

[Bru2002] Bruiltet, Sylvain. D’une mesure d’approximation simultanée à une mesure d’irrationalité : le cas de $\Gamma(1/4)$ et $\Gamma(1/3)$. Acta Arithmetica 104 (2002), no. 3, 243–281. DOI: 10.4064/aa104-3-3. PDF. Google Scholar
- [Bru2002-cor-gamma14] loc: Acta Arith. PDF p.268 (Abstract, Corollaire). quote: “$h(p/q)\ge 10^{75}\Rightarrow \lvert\Gamma(1/4)-p/q\rvert>(1/(qe))^{10^{143}}$.”
[Bru2002-def-h] loc: Acta Arith. PDF p.245 (Section 1.1, Notations et rappels). quote: “Pour $x=(x_1,\dots,x_m)\in\mathbb{Q}^m$ on d´esigne par $\lVert x\rVert_v=\max(1,\lvert x_1\rvert_v,\dots,\lvert x_m\rvert_v)$, puis $H(x)=\prod_v \lVert x\rVert_v^{d_v}$ et $h(x)=\dfrac{\log H(x)}{d(x)}$ les hauteurs relative et absolue de $x$.”
[Bru2002-def-places] loc: Acta Arith. PDF p.245 (Section 1.1, Notations et rappels). quote: “Si $K$ est un corps de nombres et $v$ une place de $K$, $\lvert\cdot\rvert_v$ d´esignera la valeur absolue normalis´ee associ´ee; on notera $d_v$ le degr´e local de $K$ en $v$.”
[Zud2004] Zudilin, Wadim. An essay on irrationality measures of $\pi$ and other logarithms. Preprint (2004). Google Scholar. arXiv PDF. arXiv abstract
- [Zud2004-def-mu] loc: arXiv PDF p.1, Definition 0.1. quote: “$\mu(\gamma)=\inf{c\in\mathbb{R}:\lvert\gamma-a/b\rvert\le \lvert b\rvert^{-c}\ \text{has finitely many solutions}}$.”
[Suk2013] Sukiennik, Justin. Bounds on Height Functions. Conference handout, 2013 Maine–Quebec Number Theory Conference (October 6, 2013). PDF
- [Suk2013-def-hpq] loc: PDF p.2, “The Height Function over a Number Field” (case $K=\mathbb{Q}$). quote: “When $K=\mathbb{Q}$, the height for $x/y\in\mathbb{Q}$ (in lowest terms) is described by $h(x/y)=\log\max{|x|,|y|}$.”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.