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.

Rational approximations to the Square root of 2.

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]

Irrationality exponent

The irrationality exponent or Liouville–Roth irrationality measure is given by setting ${\displaystyle f(q,\mu )=q^{-\mu }}$,^[1] a definition adapting the one of Liouville numbers — the irrationality exponent ${\displaystyle \mu (x)}$ is defined for real numbers ${\displaystyle x}$ to be the supremum of the set of ${\displaystyle \mu }$ such that ${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}}$ is satisfied by an infinite number of coprime integer pairs ${\displaystyle (p,q)}$ with ${\displaystyle q>0}$.^{[2][3]: 246}

For any value ${\displaystyle n<\mu (x)}$, the infinite set of all rationals ${\displaystyle p/q}$ satisfying the above inequality yields good approximations of ${\displaystyle x}$. Conversely, if ${\displaystyle n>\mu (x)}$, then there are at most finitely many coprime ${\displaystyle (p,q)}$ with ${\displaystyle q>0}$ that satisfy the inequality.

For example, whenever a rational approximation ${\displaystyle {\frac {p}{q}}\approx x}$ with ${\displaystyle p,q\in \mathbb {N} }$ yields ${\displaystyle n+1}$ exact decimal digits, then

${\displaystyle {\frac {1}{10^{n}}}\geq \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{q^{\mu (x)+\varepsilon }}}}$

for any ${\displaystyle \varepsilon >0}$, except for at most a finite number of "lucky" pairs ${\displaystyle (p,q)}$.

A number ${\displaystyle x\in \mathbb {R} }$ with irrationality exponent ${\displaystyle \mu (x)\leq 2}$ is called a diophantine number,^[4] while numbers with ${\displaystyle \mu (x)=\infty }$ 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.^[3]: 246

It is ${\displaystyle \mu (x)=\mu (rx+s)}$ for real numbers ${\displaystyle x}$ and rational numbers ${\displaystyle r\neq 0}$ and ${\displaystyle s}$. If for some ${\displaystyle x}$ we have ${\displaystyle \mu (x)\leq \mu }$, then it follows ${\displaystyle \mu (x^{1/2})\leq 2\mu }$.^[5]: 368

For a real number ${\displaystyle x}$ given by its simple continued fraction expansion ${\displaystyle x=[a_{0};a_{1},a_{2},...]}$ with convergents ${\displaystyle p_{i}/q_{i}}$ it holds:^[1]

${\displaystyle \mu (x)=1+\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{\ln q_{n}}}=2+\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{\ln q_{n}}}.}$

If we have ${\displaystyle \limsup _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}|}\leq \sigma }$ and ${\displaystyle \lim _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}x-p_{n}|}=-\tau }$ for some positive real numbers ${\displaystyle \sigma ,\tau }$, then we can establish an upper bound for the irrationality exponent of ${\displaystyle x}$ by:^[6][7]

${\displaystyle \mu (x)\leq 1+{\frac {\sigma }{\tau }}}$

Known bounds

For most transcendental numbers, the exact value of their irrationality exponent is not known.^[5] Below is a table of known upper and lower bounds.

Number ${\displaystyle x}$	Irrationality exponent ${\displaystyle \mu (x)}$		Notes
	Lower bound	Upper bound
Rational number ${\displaystyle p/q}$ with ${\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}}$	1		Every rational number ${\displaystyle p/q}$ has an irrationality exponent of exactly 1.
Irrational algebraic number ${\displaystyle \alpha }$	2		By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots and the golden ratio ${\displaystyle \varphi }$.
${\displaystyle e^{2/k},k\in \mathbb {Z} ^{+}}$	2		If the elements ${\displaystyle a_{n}}$ of the simple continued fraction expansion of an irrational number ${\displaystyle x}$ are bounded above ${\displaystyle a_{n}<P(n)}$ by an arbitrary polynomial ${\displaystyle P}$, then its irrationality exponent is ${\displaystyle \mu (x)=2}$. Examples include numbers which continued fractions behave predictably such as ${\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...]}$ and ${\displaystyle I_{0}(2)/I_{1}(2)=[1;2,3,4,5,6,7,8,9,10,...]}$.
${\displaystyle \tan(1/k),k\in \mathbb {Z} ^{+}}$	2
${\displaystyle \tanh(1/k),k\in \mathbb {Z} ^{+}}$	2
${\displaystyle S(b)}$ with ${\displaystyle b\geq 2}$	2		${\displaystyle S(b):=\sum _{k=0}^{\infty }b^{-2^{k}}}$with ${\displaystyle b\in \mathbb {Z} }$, has continued fraction terms which do not exceed a fixed constant.[8][9]
${\displaystyle T(b)}$ with ${\displaystyle b\geq 2}$[10]	2		${\displaystyle T(b):=\sum _{k=0}^{\infty }t_{k}b^{-k}}$ where ${\displaystyle t_{k}}$ is the Thue–Morse sequence and ${\displaystyle b\in \mathbb {Z} }$. See Prouhet-Thue-Morse constant.
${\displaystyle \ln(2)}$[11][12]	2	3.57455...	There are other numbers of the form ${\displaystyle \ln(a/b)}$ for which bounds on their irrationality exponents are known.[13][14][15]
${\displaystyle \ln(3)}$[11][16]	2	5.11620...
${\displaystyle 5\ln(3/2)}$[17]	2	3.43506...	There are many other numbers of the form ${\displaystyle {\sqrt {2k+1}}\ln \left({\frac {{\sqrt {2k+1}}+1}{{\sqrt {2k+1}}-1}}\right)}$ for which bounds on their irrationality exponents are known.[17] This is the case for ${\displaystyle k=12}$.
${\displaystyle \pi /{\sqrt {3}}}$[18][19]	2	4.60105...	There are many other numbers of the form ${\displaystyle {\sqrt {2k-1}}\arctan \left({\frac {\sqrt {2k-1}}{k-1}}\right)}$ for which bounds on their irrationality exponents are known.[18] This is the case for ${\displaystyle k=2}$.
${\displaystyle \pi }$[11][20]	2	7.10320...	It has been proven that if the Flint Hills series ${\displaystyle \displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}}$ (where n is in radians) converges, then ${\displaystyle \pi }$'s irrationality exponent is at most ${\displaystyle 5/2}$[21][22] and that if it diverges, the irrationality exponent is at least ${\displaystyle 5/2}$.[23]
${\displaystyle \pi ^{2}}$[11][24]	2	5.09541...	${\displaystyle \pi ^{2}}$ and ${\displaystyle \zeta (2)}$ are linearly dependent over ${\displaystyle \mathbb {Q} }$. ${\displaystyle \left(\zeta (2)={\frac {\pi ^{2}}{6}}\right)}$, also see the Basel problem.
${\displaystyle \arctan(1/2)}$[25]	2	9.27204...	There are many other numbers of the form ${\displaystyle \arctan(1/k)}$ for which bounds on their irrationality exponents are known.[26][27]
${\displaystyle \arctan(1/3)}$[28]	2	5.94202...
Apéry's constant ${\displaystyle \zeta (3)}$[11]	2	5.51389...
${\displaystyle \Gamma (1/4)}$[29]	2	10330
Cahen's constant ${\displaystyle C}$[30]	3
Champernowne constants ${\displaystyle C_{b}}$ in base ${\displaystyle b\geq 2}$[31]	${\displaystyle b}$		Examples include ${\displaystyle C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]}$
Liouville numbers ${\displaystyle L}$	${\displaystyle \infty }$		The Liouville numbers are precisely those numbers having infinite irrationality exponent.[3]: 248

Irrationality base

The irrationality base or Sondow irrationality measure is obtained by setting ${\displaystyle f(q,\beta )=\beta ^{-q}}$.^[1][6] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding ${\displaystyle \beta (x)=1}$ for all other real numbers:

Let ${\displaystyle x}$ be an irrational number. If there exist real numbers ${\displaystyle \beta \geq 1}$ with the property that for any ${\displaystyle \varepsilon >0}$, there is a positive integer ${\displaystyle q(\varepsilon )}$ such that

${\displaystyle \left|x-{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}}$

for all integers ${\displaystyle p,q}$ with ${\displaystyle q\geq q(\varepsilon )}$ then the least such ${\displaystyle \beta }$ is called the irrationality base of ${\displaystyle x}$ and is represented as ${\displaystyle \beta (x)}$.

If no such ${\displaystyle \beta }$ exists, then ${\displaystyle \beta (x)=\infty }$ and ${\displaystyle x}$ is called a super Liouville number.

If a real number ${\displaystyle x}$ is given by its simple continued fraction expansion ${\displaystyle x=[a_{0};a_{1},a_{2},...]}$ with convergents ${\displaystyle p_{i}/q_{i}}$ then it holds:

${\displaystyle \beta (x)=\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{q_{n}}}=\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{q_{n}}}}$.^[1]

Examples

Any real number ${\displaystyle x}$ with finite irrationality exponent ${\displaystyle \mu (x)<\infty }$ has irrationality base ${\displaystyle \beta (x)=1}$, while any number with irrationality base ${\displaystyle \beta (x)>1}$ has irrationality exponent ${\displaystyle \mu (x)=\infty }$ and is a Liouville number.

The number ${\displaystyle L=[1;2,2^{2},2^{2^{2}},...]}$ has irrationality exponent ${\displaystyle \mu (L)=\infty }$ and irrationality base ${\displaystyle \beta (L)=1}$.

The numbers ${\displaystyle \tau _{a}=\sum _{n=0}^{\infty }{\frac {1}{^{n}a}}=1+{\frac {1}{a}}+{\frac {1}{a^{a}}}+{\frac {1}{a^{a^{a}}}}+{\frac {1}{a^{a^{a^{a}}}}}+...}$ (${\displaystyle {^{n}a}}$ represents tetration, ${\displaystyle a=2,3,4...}$) have irrationality base ${\displaystyle \beta (\tau _{a})=a}$.

The number ${\displaystyle S=1+{\frac {1}{2^{1}}}+{\frac {1}{4^{2^{1}}}}+{\frac {1}{8^{4^{2^{1}}}}}+{\frac {1}{16^{8^{4^{2^{1}}}}}}+{\frac {1}{32^{16^{8^{4^{2^{1}}}}}}}+\ldots }$ has irrationality base ${\displaystyle \beta (S)=\infty }$, hence it is a super Liouville number.

Although it is not known whether or not ${\displaystyle e^{\pi }}$ is a Liouville number,^[32]: 20 it is known that ${\displaystyle \beta (e^{\pi })=1}$.^[5]: 371

Other irrationality measures

Markov constant

Main article: Markov constant

Setting ${\displaystyle f(q,M)=(Mq^{2})^{-1}}$ gives a stronger irrationality measure: the Markov constant ${\displaystyle M(x)}$. For an irrational number ${\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} }$ it is the factor by which Dirichlet's approximation theorem can be improved for ${\displaystyle x}$. Namely if ${\displaystyle c<M(x)}$ is a positive real number, then the inequality

${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}}$

has infinitely many solutions ${\displaystyle {\frac {p}{q}}\in \mathbb {Q} }$. If ${\displaystyle c>M(x)}$ there are at most finitely many solutions.

Dirichlet's approximation theorem implies ${\displaystyle M(x)\geq 1}$ and Hurwitz's theorem gives ${\displaystyle M(x)\geq {\sqrt {5}}}$ both for irrational ${\displaystyle x}$.^[33]

This is in fact the best general lower bound since the golden ratio gives ${\displaystyle M(\varphi )={\sqrt {5}}}$. It is also ${\displaystyle M({\sqrt {2}})=2{\sqrt {2}}}$.

Given ${\displaystyle x=[a_{0};a_{1},a_{2},...]}$ by its simple continued fraction expansion, one may obtain:^[34]

${\displaystyle M(x)=\limsup _{n\to \infty }{([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_{n},a_{n-1},...,a_{2},a_{1}])}.}$

Bounds for the Markov constant of ${\displaystyle x=[a_{0};a_{1},a_{2},...]}$ can also be given by ${\displaystyle {\sqrt {p^{2}+4}}\leq M(x)<p+2}$ with ${\displaystyle p=\limsup _{n\to \infty }a_{n}}$.^[35] This implies that ${\displaystyle M(x)=\infty }$ if and only if ${\displaystyle (a_{k})}$ is not bounded and in particular ${\displaystyle M(x)<\infty }$ if ${\displaystyle x}$ is a quadratic irrational number. A further consequence is ${\displaystyle M(e)=\infty }$.

Any number with ${\displaystyle \mu (x)>2}$ or ${\displaystyle \beta (x)>1}$ has an unbounded simple continued fraction and hence ${\displaystyle M(x)=\infty }$.

For rational numbers ${\displaystyle r}$ it may be defined ${\displaystyle M(r)=0}$.

Other results

The values ${\displaystyle M(e)=\infty }$ and ${\displaystyle \mu (e)=2}$ imply that the inequality ${\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}}$ has for all ${\displaystyle c\in \mathbb {R} ^{+}}$ infinitely many solutions ${\displaystyle {\frac {p}{q}}\in \mathbb {Q} }$ while the inequality ${\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{q^{2+\varepsilon }}}}$ has for all ${\displaystyle \varepsilon \in \mathbb {R} ^{+}}$ only at most finitely many solutions ${\displaystyle {\frac {p}{q}}\in \mathbb {Q} }$ . This gives rise to the question what the best upper bound is. The answer is given by:^[36]

${\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {c\ln \ln q}{q^{2}\ln q}}}$

which is satisfied by infinitely many ${\displaystyle {\frac {p}{q}}\in \mathbb {Q} }$ for ${\displaystyle c>{\tfrac {1}{2}}}$ but not for ${\displaystyle c<{\tfrac {1}{2}}}$.

This makes the number ${\displaystyle e}$ alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers ${\displaystyle x\in \mathbb {R} }$ the inequality below has infinitely many solutions ${\displaystyle {\frac {p}{q}}\in \mathbb {Q} }$:^[5] (see Khinchin's theorem)

${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{2}\ln q}}}$

Mahler's generalization

Main article: Transcendental number theory § Mahler's_classification

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.^[3]

Mahler's irrationality measure

Instead of taking for a given real number ${\displaystyle x}$ the difference ${\displaystyle |x-p/q|}$ with ${\displaystyle p/q\in \mathbb {Q} }$, one may instead focus on term ${\displaystyle |qx-p|=|L(x)|}$ with ${\displaystyle p,q\in \mathbb {Z} }$ and ${\displaystyle L\in \mathbb {Z} [x]}$ with ${\displaystyle \deg L=1}$. Consider the following inequality:

${\displaystyle 0<|qx-p|\leq \max(|p|,|q|)^{-\omega }}$ with ${\displaystyle p,q\in \mathbb {Z} }$ and ${\displaystyle \omega \in \mathbb {R} _{0}^{+}}$.

Define ${\displaystyle R}$ as the set of all ${\displaystyle \omega \in \mathbb {R} _{0}^{+}}$ for which infinitely many solutions ${\displaystyle p,q\in \mathbb {Z} }$ exist, such that the inequality is satisfied. Then ${\displaystyle \omega _{1}(x)=\sup M}$ is Mahler's irrationality measure. It gives ${\displaystyle \omega _{1}(p/q)=0}$ for rational numbers, ${\displaystyle \omega _{1}(\alpha )=1}$ for algebraic irrational numbers and in general ${\displaystyle \omega _{1}(x)=\mu (x)-1}$, where ${\displaystyle \mu (x)}$ denotes the irrationality exponent.

Transcendence measure

Mahler's irrationality measure can be generalized as follows:^[2][3] Take ${\displaystyle P}$ to be a polynomial with ${\displaystyle \deg P\leq n\in \mathbb {Z} ^{+}}$ and integer coefficients ${\displaystyle a_{i}\in \mathbb {Z} }$. Then define a height function ${\displaystyle H(P)=\max(|a_{0}|,|a_{1}|,...,|a_{n}|)}$ and consider for complex numbers ${\displaystyle z}$ the inequality:

${\displaystyle 0<|P(z)|\leq H(P)^{-\omega }}$ with ${\displaystyle \omega \in \mathbb {R} _{0}^{+}}$.

Set ${\displaystyle R}$ to be the set of all ${\displaystyle \omega \in \mathbb {R} _{0}^{+}}$ for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define ${\displaystyle \omega _{n}(z)=\sup R}$ for all ${\displaystyle n\in \mathbb {Z} ^{+}}$ with ${\displaystyle \omega _{1}(z)}$ being the above irrationality measure, ${\displaystyle \omega _{2}(z)}$ being a non-quadraticity measure, etc.

Then Mahler's transcendence measure is given by:

${\displaystyle \omega (z)=\limsup _{n\to \infty }\omega _{n}(z).}$

The transcendental numbers can now be divided into the following three classes:

If for all ${\displaystyle n\in \mathbb {Z} ^{+}}$ the value of ${\displaystyle \omega _{n}(z)}$ is finite and ${\displaystyle \omega (z)}$ is finite as well, ${\displaystyle z}$ is called an S-number (of type ${\displaystyle \omega (z)}$).

If for all ${\displaystyle n\in \mathbb {Z} ^{+}}$ the value of ${\displaystyle \omega _{n}(z)}$ is finite but ${\displaystyle \omega (z)}$ is infinite, ${\displaystyle z}$ is called an T-number.

If there exists a smallest positive integer ${\displaystyle N}$ such that for all ${\displaystyle n\geq N}$ the ${\displaystyle \omega _{n}(z)}$ are infinite, ${\displaystyle z}$ is called an U-number (of degree ${\displaystyle N}$).

The number ${\displaystyle z}$ is algebraic (and called an A-number) if and only if ${\displaystyle \omega (z)=0}$.

Almost all numbers are S-numbers. In fact, almost all real numbers give ${\displaystyle \omega (x)=1}$ while almost all complex numbers give ${\displaystyle \omega (z)={\tfrac {1}{2}}}$.^[37]: 86 The number e is an S-number with ${\displaystyle \omega (e)=1}$. The number π is either an S- or T-number.^[37]: 86 The U-numbers are a set of measure 0 but still uncountable.^[38] 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.^[2][13] For real numbers ${\displaystyle x_{1},...,x_{n}\in \mathbb {R} }$ consider the inequality

${\displaystyle 0<|c_{1}x_{1}+...+c_{n}x_{n}|\leq \max(|c_{1}|,...,|c_{n}|)^{-\nu }}$ with ${\displaystyle c_{1},...,c_{n}\in \mathbb {Z} }$ and ${\displaystyle \nu \in \mathbb {R} _{0}^{+}}$.

Define ${\displaystyle R}$ as the set of all ${\displaystyle \nu \in \mathbb {R} _{0}^{+}}$ for which infinitely many solutions ${\displaystyle c_{1},...c_{n}\in \mathbb {Z} }$ exist, such that the inequality is satisfied. Then ${\displaystyle \nu (x_{1},...,x_{n})=\sup R}$ is the linear independence measure.

If the ${\displaystyle x_{1},...,x_{n}}$ are linearly dependent over ${\displaystyle \mathbb {\mathbb {Q} } }$ then ${\displaystyle \nu (x_{1},...,x_{n})=0}$.

If ${\displaystyle 1,x_{1},...,x_{n}}$ are linearly independent algebraic numbers over ${\displaystyle \mathbb {\mathbb {Q} } }$ then ${\displaystyle \nu (1,x_{1},...,x_{n})\leq n}$.^[32]

It is further ${\displaystyle \nu (1,x)=\omega _{1}(x)=\mu (x)-1}$.

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.^[3][37]

For a given complex number ${\displaystyle z}$ consider algebraic numbers ${\displaystyle \alpha }$ of degree at most ${\displaystyle n}$. Define a height function ${\displaystyle H(\alpha )=H(P)}$, where ${\displaystyle P}$ is the characteristic polynomial of ${\displaystyle \alpha }$ and consider the inequality:

${\displaystyle 0<|z-\alpha |\leq H(\alpha )^{-\omega ^{*}-1}}$ with ${\displaystyle \omega ^{*}\in \mathbb {R} _{0}^{+}}$.

Set ${\displaystyle R}$ to be the set of all ${\displaystyle \omega ^{*}\in \mathbb {R} _{0}^{+}}$ for which infinitely many such algebraic numbers ${\displaystyle \alpha }$ exist, that keep the inequality satisfied. Further define ${\displaystyle \omega _{n}^{*}(z)=\sup R}$ for all ${\displaystyle n\in \mathbb {Z} ^{+}}$ with ${\displaystyle \omega _{1}^{*}(z)}$ being an irrationality measure, ${\displaystyle \omega _{2}^{*}(z)}$ being a non-quadraticity measure,^[17] etc.

Then Koksma's transcendence measure is given by:

${\displaystyle \omega ^{*}(z)=\limsup _{n\to \infty }\omega _{n}^{*}(z)}$.

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.^[37]: 87

Simultaneous approximation of real numbers

Main article: Subspace theorem

Given a real number ${\displaystyle x\in \mathbb {R} }$, an irrationality measure of ${\displaystyle x}$ quantifies how well it can be approximated by rational numbers ${\displaystyle {\frac {p}{q}}}$ with denominator ${\displaystyle q\in \mathbb {Z} ^{+}}$. If ${\displaystyle x=\alpha }$ is taken to be an algebraic number that is also irrational one may obtain that the inequality

${\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}}$

has only at most finitely many solutions ${\displaystyle {\frac {p}{q}}\in \mathbb {Q} }$ for ${\displaystyle \mu >2}$. This is known as Roth's theorem.

This can be generalized: Given a set of real numbers ${\displaystyle x_{1},...,x_{n}\in \mathbb {R} }$ one can quantify how well they can be approximated simultaneously by rational numbers ${\displaystyle {\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}}$ with the same denominator ${\displaystyle q\in \mathbb {Z} ^{+}}$. If the ${\displaystyle x_{i}=\alpha _{i}}$ are taken to be algebraic numbers, such that ${\displaystyle 1,\alpha _{1},...,\alpha _{n}}$ are linearly independent over the rational numbers ${\displaystyle \mathbb {Q} }$ it follows that the inequalities

${\displaystyle 0<\left|\alpha _{i}-{\frac {p_{i}}{q}}\right|<{\frac {1}{q^{\mu }}},\forall i\in \{1,...,n\}}$

have only at most finitely many solutions ${\displaystyle \left({\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}\right)\in \mathbb {Q} ^{n}}$ for ${\displaystyle \mu >1+{\frac {1}{n}}}$. This result is due to Wolfgang M. Schmidt.^[39][40]

See also

• Diophantine approximation
• Transcendental number
• Continued fraction
• Brjuno number