# Irrationality measure

In mathematics, an irrationality measure of a real number $x$ is a measure of how "closely" it can be approximated by rationals.
If a function $f(t,\lambda )$, defined for $t,\lambda >0$, takes positive real values and is strictly decreasing in both variables, consider the following inequality:
    $0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda )$
for a given real number $x\in \mathbb {R}$ and rational numbers ${\frac {p}{q}}$ with $p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}$. Define $R$ as the set of all $\lambda \in \mathbb {R} ^{+}$ for which only finitely many ${\frac {p}{q}}$ exist, such that the inequality is satisfied. Then $\lambda (x)=\inf R$ is called an irrationality measure of $x$ with regard to $f.$ If there is no such $\lambda$ and the set $R$ is empty, $x$ is said to have infinite irrationality measure $\lambda (x)=\infty$.
Consequently, the inequality
    $0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda (x)+\varepsilon )$
has at most only finitely many solutions ${\frac {p}{q}}$ for all $\varepsilon >0$.

## Irrationality exponent

The irrationality exponent or Liouville–Roth irrationality measure is given by setting $f(q,\mu )=q^{-\mu }$, a definition adapting the one of Liouville numbers — the irrationality exponent $\mu (x)$ is defined for real numbers $x$ to be the supremum of the set of $\mu$ such that $0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}$ is satisfied by an infinite number of coprime integer pairs $(p,q)$ with $q>0$.$^{:246}$
For any value $n<\mu (x)$, the infinite set of all rationals $p/q$ satisfying the above inequality yields good approximations of $x$. Conversely, if $n>\mu (x)$, then there are at most finitely many coprime $(p,q)$ with $q>0$ that satisfy the inequality.
For example, whenever a rational approximation ${\frac {p}{q}}\approx x$ with $p,q\in \mathbb {N}$ yields $n+1$ exact decimal digits, then
    ${\frac {1}{10^{n}}}\geq \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{q^{\mu (x)+\varepsilon }}}$
for any $\varepsilon >0$, except for at most a finite number of "lucky" pairs $(p,q)$.
A number $x\in \mathbb {R}$ with irrationality exponent $\mu (x)\leq 2$ is called a diophantine number, while numbers with $\mu (x)=\infty$ are called Liouville numbers.

### Corollaries

Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2.
On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers, have an irrationality exponent exactly equal to 2.$^{:246}$
It is $\mu (x)=\mu (rx+s)$ for real numbers $x$ and rational numbers $r\neq 0$ and $s$. If for some $x$ we have $\mu (x)\leq \mu$, then it follows $\mu (x^{1/2})\leq 2\mu$.$^{:368}$
For a real number $x$ given by its simple continued fraction expansion $x=[a_{0};a_{1},a_{2},...]$ with convergents $p_{i}/q_{i}$ it holds:
    $\mu (x)=1+\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{\ln q_{n}}}=2+\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{\ln q_{n}}}.$
If we have $\limsup _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}|}\leq \sigma$ and $\lim _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}x-p_{n}|}=-\tau$ for some positive real numbers $\sigma ,\tau$, then we can establish an upper bound for the irrationality exponent of $x$ by:
    $\mu (x)\leq 1+{\frac {\sigma }{\tau }}$

### Known bounds

For most transcendental numbers, the exact value of their irrationality exponent is not known. Below is a table of known upper and lower bounds.
| Number $x$ | Irrationality exponent $\mu (x)$ | Irrationality exponent $\mu (x)$ | Notes |
| Number $x$ | Lower bound | Upper bound | Notes |
| Rational number $p/q$ with $p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}$ | 1 | 1 | Every rational number $p/q$ has an irrationality exponent of exactly 1. |
| Irrational algebraic number $\alpha$ | 2 | 2 | By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots and the golden ratio $\varphi$. |
| $e^{2/k},k\in \mathbb {Z} ^{+}$ | 2 | 2 | If the elements $a_{n}$ of the simple continued fraction expansion of an irrational number $x$ are bounded above $a_{n}<P(n)$ by an arbitrary polynomial $P$, then its irrationality exponent is $\mu (x)=2$. Examples include numbers which continued fractions behave predictably such as $e=[2;1,2,1,1,4,1,1,6,1,...]$ and $I_{0}(2)/I_{1}(2)=[1;2,3,4,5,6,7,8,9,10,...]$. |
| $\tan(1/k),k\in \mathbb {Z} ^{+}$ | 2 | 2 | | If the elements $a_{n}$ of the simple continued fraction expansion of an irrational number $x$ are bounded above $a_{n}<P(n)$ by an arbitrary polynomial $P$, then its irrationality exponent is $\mu (x)=2$. Examples include numbers which continued fractions behave predictably such as $e=[2;1,2,1,1,4,1,1,6,1,...]$ and $I_{0}(2)/I_{1}(2)=[1;2,3,4,5,6,7,8,9,10,...]$. |
| $\tanh(1/k),k\in \mathbb {Z} ^{+}$ | 2 | 2 | | If the elements $a_{n}$ of the simple continued fraction expansion of an irrational number $x$ are bounded above $a_{n}<P(n)$ by an arbitrary polynomial $P$, then its irrationality exponent is $\mu (x)=2$. Examples include numbers which continued fractions behave predictably such as $e=[2;1,2,1,1,4,1,1,6,1,...]$ and $I_{0}(2)/I_{1}(2)=[1;2,3,4,5,6,7,8,9,10,...]$. |
| $S(b)$ with $b\geq 2$ | 2 | 2 | $S(b):=\sum _{k=0}^{\infty }b^{-2^{k}}$with $b\in \mathbb {Z}$, has continued fraction terms which do not exceed a fixed constant. |
| $T(b)$ with $b\geq 2$ | 2 | 2 | $T(b):=\sum _{k=0}^{\infty }t_{k}b^{-k}$ where $t_{k}$ is the Thue–Morse sequence and $b\in \mathbb {Z}$. See Prouhet-Thue-Morse constant. |
| $\ln(2)$ | 2 | 3.57455... | There are other numbers of the form $\ln(a/b)$ for which bounds on their irrationality exponents are known. |
| $\ln(3)$ | 2 | 5.11620... | There are other numbers of the form $\ln(a/b)$ for which bounds on their irrationality exponents are known. |
| $5\ln(3/2)$ | 2 | 3.43506... | There are many other numbers of the form ${\sqrt {2k+1}}\ln \left({\frac {{\sqrt {2k+1}}+1}{{\sqrt {2k+1}}-1}}\right)$ for which bounds on their irrationality exponents are known. This is the case for $k=12$. |
| $\pi /{\sqrt {3}}$ | 2 | 4.60105... | There are many other numbers of the form ${\sqrt {2k-1}}\arctan \left({\frac {\sqrt {2k-1}}{k-1}}\right)$ for which bounds on their irrationality exponents are known. This is the case for $k=2$. |
| $\pi$ | 2 | 7.10320... | It has been proven that if the Flint Hills series $\displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}$ (where n is in radians) converges, then $\pi$'s irrationality exponent is at most $5/2$ and that if it diverges, the irrationality exponent is at least $5/2$. |
| $\pi ^{2}$ | 2 | 5.09541... | $\pi ^{2}$ and $\zeta (2)$ are linearly dependent over $\mathbb {Q}$. $\left(\zeta (2)={\frac {\pi ^{2}}{6}}\right)$, also see the Basel problem. |
| $\arctan(1/2)$ | 2 | 9.27204... | There are many other numbers of the form $\arctan(1/k)$ for which bounds on their irrationality exponents are known. |
| $\arctan(1/3)$ | 2 | 5.94202... | There are many other numbers of the form $\arctan(1/k)$ for which bounds on their irrationality exponents are known. |
| Apéry's constant $\zeta (3)$ | 2 | 5.51389... | |
| $\Gamma (1/4)$ | 2 | 10$^{330}$ | |
| Cahen's constant $C$ | 3 | 3 | |
| Champernowne constants $C_{b}$ in base $b\geq 2$ | $b$ | $b$ | Examples include $C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]$ |
| Liouville numbers $L$ | $\infty$ | $\infty$ | The Liouville numbers are precisely those numbers having infinite irrationality exponent.$^{:248}$ |

## Irrationality base

The irrationality base or Sondow irrationality measure is obtained by setting $f(q,\beta )=\beta ^{-q}$. It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding $\beta (x)=1$ for all other real numbers:
Let $x$ be an irrational number. If there exist real numbers $\beta \geq 1$ with the property that for any $\varepsilon >0$, there is a positive integer $q(\varepsilon )$ such that
    $\left|x-{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}$
for all integers $p,q$ with $q\geq q(\varepsilon )$ then the least such $\beta$ is called the irrationality base of $x$ and is represented as $\beta (x)$.
If no such $\beta$ exists, then $\beta (x)=\infty$ and $x$ is called a super Liouville number.
If a real number $x$ is given by its simple continued fraction expansion $x=[a_{0};a_{1},a_{2},...]$ with convergents $p_{i}/q_{i}$ then it holds:
    $\beta (x)=\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{q_{n}}}=\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{q_{n}}}$.

### Examples

Any real number $x$ with finite irrationality exponent $\mu (x)<\infty$ has irrationality base $\beta (x)=1$, while any number with irrationality base $\beta (x)>1$ has irrationality exponent $\mu (x)=\infty$ and is a Liouville number.
The number $L=[1;2,2^{2},2^{2^{2}},...]$ has irrationality exponent $\mu (L)=\infty$ and irrationality base $\beta (L)=1$.
The numbers $\tau _{a}=\sum _{n=0}^{\infty }{\frac {1}{^{n}a}}=1+{\frac {1}{a}}+{\frac {1}{a^{a}}}+{\frac {1}{a^{a^{a}}}}+{\frac {1}{a^{a^{a^{a}}}}}+...$ (${^{n}a}$ represents tetration, $a=2,3,4...$) have irrationality base $\beta (\tau _{a})=a$.
The number $S=1+{\frac {1}{2^{1}}}+{\frac {1}{4^{2^{1}}}}+{\frac {1}{8^{4^{2^{1}}}}}+{\frac {1}{16^{8^{4^{2^{1}}}}}}+{\frac {1}{32^{16^{8^{4^{2^{1}}}}}}}+\ldots$ has irrationality base $\beta (S)=\infty$, hence it is a super Liouville number.
Although it is not known whether or not $e^{\pi }$ is a Liouville number,$^{:20}$ it is known that $\beta (e^{\pi })=1$.$^{:371}$

## Other irrationality measures
### Markov constant

Setting $f(q,M)=(Mq^{2})^{-1}$ gives a stronger irrationality measure: the Markov constant $M(x)$. For an irrational number $x\in \mathbb {R} \setminus \mathbb {Q}$ it is the factor by which Dirichlet's approximation theorem can be improved for $x$. Namely if $c<M(x)$ is a positive real number, then the inequality
    $0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}$
has infinitely many solutions ${\frac {p}{q}}\in \mathbb {Q}$. If $c>M(x)$ there are at most finitely many solutions.
Dirichlet's approximation theorem implies $M(x)\geq 1$ and Hurwitz's theorem gives $M(x)\geq {\sqrt {5}}$ both for irrational $x$.
This is in fact the best general lower bound since the golden ratio gives $M(\varphi )={\sqrt {5}}$. It is also $M({\sqrt {2}})=2{\sqrt {2}}$.
Given $x=[a_{0};a_{1},a_{2},...]$ by its simple continued fraction expansion, one may obtain:
    $M(x)=\limsup _{n\to \infty }{([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_{n},a_{n-1},...,a_{2},a_{1}])}.$
Bounds for the Markov constant of $x=[a_{0};a_{1},a_{2},...]$ can also be given by ${\sqrt {p^{2}+4}}\leq M(x)<p+2$ with $p=\limsup _{n\to \infty }a_{n}$. This implies that $M(x)=\infty$ if and only if $(a_{k})$ is not bounded and in particular $M(x)<\infty$ if $x$ is a quadratic irrational number. A further consequence is $M(e)=\infty$.
Any number with $\mu (x)>2$ or $\beta (x)>1$ has an unbounded simple continued fraction and hence $M(x)=\infty$.
For rational numbers $r$ it may be defined $M(r)=0$.

### Other results

The values $M(e)=\infty$ and $\mu (e)=2$ imply that the inequality $0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}$ has for all $c\in \mathbb {R} ^{+}$ infinitely many solutions ${\frac {p}{q}}\in \mathbb {Q}$ while the inequality $0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{q^{2+\varepsilon }}}$ has for all $\varepsilon \in \mathbb {R} ^{+}$ only at most finitely many solutions ${\frac {p}{q}}\in \mathbb {Q}$ . This gives rise to the question what the best upper bound is. The answer is given by:
    $0<\left|e-{\frac {p}{q}}\right|<{\frac {c\ln \ln q}{q^{2}\ln q}}$
which is satisfied by infinitely many ${\frac {p}{q}}\in \mathbb {Q}$ for $c>{\tfrac {1}{2}}$ but not for $c<{\tfrac {1}{2}}$.
This makes the number $e$ alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers $x\in \mathbb {R}$ the inequality below has infinitely many solutions ${\frac {p}{q}}\in \mathbb {Q}$: (see Khinchin's theorem)
    $0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{2}\ln q}}$

## Mahler's generalization

Kurt Mahler extended the concept of an irrationality measure and defined a so-called transcendence measure, drawing on the idea of a Liouville number and partitioning the transcendental numbers into three distinct classes.

### Mahler's irrationality measure

Instead of taking for a given real number $x$ the difference $|x-p/q|$ with $p/q\in \mathbb {Q}$, one may instead focus on term $|qx-p|=|L(x)|$ with $p,q\in \mathbb {Z}$ and $L\in \mathbb {Z} [x]$ with $\deg L=1$. Consider the following inequality:
$0<|qx-p|\leq \max(|p|,|q|)^{-\omega }$ with $p,q\in \mathbb {Z}$ and $\omega \in \mathbb {R} _{0}^{+}$.
Define $R$ as the set of all $\omega \in \mathbb {R} _{0}^{+}$ for which infinitely many solutions $p,q\in \mathbb {Z}$ exist, such that the inequality is satisfied. Then $\omega _{1}(x)=\sup M$ is Mahler's irrationality measure. It gives $\omega _{1}(p/q)=0$ for rational numbers, $\omega _{1}(\alpha )=1$ for algebraic irrational numbers and in general $\omega _{1}(x)=\mu (x)-1$, where $\mu (x)$ denotes the irrationality exponent.

### Transcendence measure

Mahler's irrationality measure can be generalized as follows: Take $P$ to be a polynomial with $\deg P\leq n\in \mathbb {Z} ^{+}$ and integer coefficients $a_{i}\in \mathbb {Z}$. Then define a height function $H(P)=\max(|a_{0}|,|a_{1}|,...,|a_{n}|)$ and consider for complex numbers $z$ the inequality:
$0<|P(z)|\leq H(P)^{-\omega }$ with $\omega \in \mathbb {R} _{0}^{+}$.
Set $R$ to be the set of all $\omega \in \mathbb {R} _{0}^{+}$ for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define $\omega _{n}(z)=\sup R$ for all $n\in \mathbb {Z} ^{+}$ with $\omega _{1}(z)$ being the above irrationality measure, $\omega _{2}(z)$ being a non-quadraticity measure, etc.
Then Mahler's transcendence measure is given by:
    $\omega (z)=\limsup _{n\to \infty }\omega _{n}(z).$
The transcendental numbers can now be divided into the following three classes:
If for all $n\in \mathbb {Z} ^{+}$ the value of $\omega _{n}(z)$ is finite and $\omega (z)$ is finite as well, $z$ is called an S-number (of type $\omega (z)$).
If for all $n\in \mathbb {Z} ^{+}$ the value of $\omega _{n}(z)$ is finite but $\omega (z)$ is infinite, $z$ is called an T-number.
If there exists a smallest positive integer $N$ such that for all $n\geq N$ the $\omega _{n}(z)$ are infinite, $z$ is called an U-number (of degree $N$).
The number $z$ is algebraic (and called an A-number) if and only if $\omega (z)=0$.
Almost all numbers are S-numbers. In fact, almost all real numbers give $\omega (x)=1$ while almost all complex numbers give $\omega (z)={\tfrac {1}{2}}$.$^{:86}$ The number e is an S-number with $\omega (e)=1$. The number π is either an S- or T-number.$^{:86}$ The U-numbers are a set of measure 0 but still uncountable. They contain the Liouville numbers which are exactly the U-numbers of degree one.

### Linear independence measure

Another generalization of Mahler's irrationality measure gives a linear independence measure. For real numbers $x_{1},...,x_{n}\in \mathbb {R}$ consider the inequality
$0<|c_{1}x_{1}+...+c_{n}x_{n}|\leq \max(|c_{1}|,...,|c_{n}|)^{-\nu }$ with $c_{1},...,c_{n}\in \mathbb {Z}$ and $\nu \in \mathbb {R} _{0}^{+}$.
Define $R$ as the set of all $\nu \in \mathbb {R} _{0}^{+}$ for which infinitely many solutions $c_{1},...c_{n}\in \mathbb {Z}$ exist, such that the inequality is satisfied. Then $\nu (x_{1},...,x_{n})=\sup R$ is the linear independence measure.
If the $x_{1},...,x_{n}$ are linearly dependent over $\mathbb {\mathbb {Q} }$ then $\nu (x_{1},...,x_{n})=0$.
If $1,x_{1},...,x_{n}$ are linearly independent algebraic numbers over $\mathbb {\mathbb {Q} }$ then $\nu (1,x_{1},...,x_{n})\leq n$.
It is further $\nu (1,x)=\omega _{1}(x)=\mu (x)-1$.

## Other generalizations
### Koksma’s generalization

Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of complex numbers by algebraic numbers.
For a given complex number $z$ consider algebraic numbers $\alpha$ of degree at most $n$. Define a height function $H(\alpha )=H(P)$, where $P$ is the characteristic polynomial of $\alpha$ and consider the inequality:
$0<|z-\alpha |\leq H(\alpha )^{-\omega ^{*}-1}$ with $\omega ^{*}\in \mathbb {R} _{0}^{+}$.
Set $R$ to be the set of all $\omega ^{*}\in \mathbb {R} _{0}^{+}$ for which infinitely many such algebraic numbers $\alpha$ exist, that keep the inequality satisfied. Further define $\omega _{n}^{*}(z)=\sup R$ for all $n\in \mathbb {Z} ^{+}$ with $\omega _{1}^{*}(z)$ being an irrationality measure, $\omega _{2}^{*}(z)$ being a non-quadraticity measure, etc.
Then Koksma's transcendence measure is given by:
    $\omega ^{*}(z)=\limsup _{n\to \infty }\omega _{n}^{*}(z)$.
The complex numbers can now once again be partitioned into four classes A*, S*, T* and U*. However it turns out that these classes are equivalent to the ones given by Mahler in the sense that they produce exactly the same partition.$^{:87}$

### Simultaneous approximation of real numbers

Given a real number $x\in \mathbb {R}$, an irrationality measure of $x$ quantifies how well it can be approximated by rational numbers ${\frac {p}{q}}$ with denominator $q\in \mathbb {Z} ^{+}$. If $x=\alpha$ is taken to be an algebraic number that is also irrational one may obtain that the inequality
    $0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}$
has only at most finitely many solutions ${\frac {p}{q}}\in \mathbb {Q}$ for $\mu >2$. This is known as Roth's theorem.
This can be generalized: Given a set of real numbers $x_{1},...,x_{n}\in \mathbb {R}$ one can quantify how well they can be approximated simultaneously by rational numbers ${\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}$ with the same denominator $q\in \mathbb {Z} ^{+}$. If the $x_{i}=\alpha _{i}$ are taken to be algebraic numbers, such that $1,\alpha _{1},...,\alpha _{n}$ are linearly independent over the rational numbers $\mathbb {Q}$ it follows that the inequalities
    $0<\left|\alpha _{i}-{\frac {p_{i}}{q}}\right|<{\frac {1}{q^{\mu }}},\forall i\in \{1,...,n\}$
have only at most finitely many solutions $\left({\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}\right)\in \mathbb {Q} ^{n}$ for $\mu >1+{\frac {1}{n}}$. This result is due to Wolfgang M. Schmidt.