Irrationality measure

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In mathematics, an irrationality measure of a real number x {\displaystyle x} is a measure of how closely it can be approximated by a rational number. If a function f ( t , λ ) {\displaystyle f(t,\lambda )} , defined for t , λ > 0 {\displaystyle t,\lambda >0} , takes positive real values and is strictly decreasing in both variables, consider the following inequality: 0 < | x − p q | < f ( q , λ ) {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda )} for a given real number x ∈ R {\displaystyle x\in \mathbb {R} } and rational numbers p q {\displaystyle {\frac {p}{q}}} with p ∈ Z , q ∈ Z + {\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}} . For x ∈ R {\displaystyle x\in \mathbb {R} } , define R {\displaystyle R} as the set of all λ ∈ R + {\displaystyle \lambda \in \mathbb {R} ^{+}} for which the set { p q ∈ Q : | x − p q | < f ( q , λ ) } {\displaystyle \left\{{\frac {p}{q}}\in \mathbb {Q} :\left\vert x-{\frac {p}{q}}\right\vert <f(q,\lambda )\right\}} is finite. Then λ ( x ) = inf R {\displaystyle \lambda (x)=\inf R} is called an irrationality measure of x {\displaystyle x} with regard to f . {\displaystyle f.} If the set R {\displaystyle R} is empty, x {\displaystyle x} is said to have infinite irrationality measure λ ( x ) = ∞ {\displaystyle \lambda (x)=\infty } . By definition, there is a sequence ( ε n ) n ∈ N {\displaystyle (\varepsilon _{n})_{n\in \mathbb {N} }} converging to 0 such that for every n {\displaystyle n} , the inequality 0 < | x − p q | < f ( q , λ ( x ) + ε n ) {\displaystyle 0<\left|x-{\frac...