Irrationality measure

In mathematics, an irrationality measure of a real number ${\displaystyle x}$ is a measure of how "closely" it can be approximated by rationals.

If a function ${\displaystyle f(t,\lambda )}$, defined for ${\displaystyle t,\lambda >0}$, takes positive real values and is strictly decreasing in both variables, consider the following inequality:

${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda )}$

for a given real number ${\displaystyle x\in \mathbb {R} }$ and rational numbers ${\displaystyle {\frac {p}{q}}}$ with ${\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}}$. Define ${\displaystyle R}$ as the set of all ${\displaystyle \lambda \in \mathbb {R} ^{+}}$ for which only finitely many ${\displaystyle {\frac {p}{q}}}$ exist, such that the inequality is satisfied. Then ${\displaystyle \lambda (x)=\inf R}$ is called an irrationality measure of ${\displaystyle x}$ with regard to ${\displaystyle f.}$ If there is no such ${\displaystyle \lambda }$ and the set ${\displaystyle R}$ is empty, ${\displaystyle x}$ is said to have infinite irrationality measure ${\displaystyle \lambda (x)=\infty }$.

Consequently the inequality

${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda (x)+\varepsilon )}$

has at most only finitely many solutions ${\displaystyle {\frac {p}{q}}}$ for all ${\displaystyle \varepsilon >0}$.^[1]