Irrationality measure

Rational approximations to the Square root of 2.

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:

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

for a given real number $x\in ℝ$ and rational numbers $\frac{p}{q}$ with $p\in \mathbb{Z},q\in {\mathbb{Z}}^{+}$. Define $R$ as the set of all $\lambda \in {ℝ}^{+}$ 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 $\lambda (x)=\mathrm{\infty }$.

Consequently, the inequality

$0<|x-\frac{p}{q}|<f(q,\lambda (x)+\epsilon )$

has at most only finitely many solutions $\frac{p}{q}$ for all $\epsilon >0$.

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 $0<|x-\frac{p}{q}|<\frac{1}{q^{\mu }}$ is satisfied by an infinite number of coprime integer pairs $(p,q)$ with $q>0$.Template:Rp

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

$\frac{1}{10^n}\ge |x-\frac{p}{q}|\ge \frac{1}{q^{\mu (x)+\epsilon }}$

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

A number $x\in ℝ$ with irrationality exponent $\mu (x)\le 2$ is called a diophantine number, while numbers with $\mu (x)=\mathrm{\infty }$ 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.Template:Rp

It is $\mu (x)=\mu (rx+s)$ for real numbers $x$ and rational numbers $r\ne 0$ and $s$. If for some $x$ we have $\mu (x)\le \mu$, then it follows $\mu (x^{1/2})\le 2\mu$.Template:Rp

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:

$\mu (x)=1+\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{\ln q_{n+1}}{\ln q_n}=2+\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{\ln a_{n+1}}{\ln q_n}.$

If we have $\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{1}{n}\ln |q_n|\le \sigma$ and $\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\ln |q_{n}x-p_n|=-\tau$ for some positive real numbers $\sigma ,\tau$, then we can establish an upper bound for the irrationality exponent of $x$ by:

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

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 $\mu (x)$		Notes
	Lower bound	Upper bound
Rational number $p/q$ with $p\in \mathbb{Z},q\in {\mathbb{Z}}^{+}$	1		Every rational number $p/q$ has an irrationality exponent of exactly 1.
Irrational algebraic number $\alpha$	2		By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots and the golden ratio $\phi$.
$e^{2/k},k\in {\mathbb{Z}}^{+}$	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
$\tanh (1/k),k\in {\mathbb{Z}}^{+}$	2
$S(b)$ with $b\ge 2$	2		$S(b):=\sum _{k=0}^{\mathrm{\infty }}{b}^{-2^k}$with $b\in \mathbb{Z}$, has continued fraction terms which do not exceed a fixed constant.
$T(b)$ with $b\ge 2$	2		$T(b):=\sum _{k=0}^{\mathrm{\infty }}{t}_k{b}^{-k}$ 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...
$5\ln (3/2)$	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 $\sum _{n=1}^{\mathrm{\infty }}\frac{{\csc }^{2}n}{n^3}$ (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 $\mathbb{Q}$. $(\zeta (2)=\frac{{\pi }^2}{6})$, 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...
Apéry's constant $\zeta (3)$	2	5.51389...
$\mathrm{\Gamma }(1/4)$	2	10330
Cahen's constant $C$	3
Champernowne constants $C_b$ in base $b\ge 2$	$b$		Examples include $C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]$
Liouville numbers $L$	$\mathrm{\infty }$		The Liouville numbers are precisely those numbers having infinite irrationality exponent.Template:Rp

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 $\beta \ge 1$ with the property that for any $\epsilon >0$, there is a positive integer $q(\epsilon )$ such that

$|x-\frac{p}{q}|>\frac{1}{(\beta +\epsilon {)}^q}$

for all integers $p,q$ with $q\ge q(\epsilon )$ then the least such $\beta$ is called the irrationality base of $x$ and is represented as $\beta (x)$.

If no such $\beta$ exists, then $\beta (x)=\mathrm{\infty }$ 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:

$\beta (x)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{\ln q_{n+1}}{q_n}=\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{\ln a_{n+1}}{q_n}$.

Examples

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

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

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

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

Although it is not known whether or not $e^{\pi }$ is a Liouville number,Template:Rp it is known that $\beta (e^{\pi })=1$.Template:Rp

Other irrationality measures

Markov constant

Template:Main Setting $f(q,M)=(M{q}^2{)}^{-1}$ gives a stronger irrationality measure: the Markov constant $M(x)$. For an irrational number $x\in ℝ\setminus \mathbb{Q}$ 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

$0<|x-\frac{p}{q}|<\frac{1}{c{q}^2}$

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 $M(x)\ge 1$ and Hurwitz's theorem gives $M(x)\ge \sqrt{5}$ both for irrational $x$.

This is in fact the best general lower bound since the golden ratio gives $M(\phi )=\sqrt{5}$. 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:

$M(x)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_n,a_{n-1},...,a_2,a_1]).$

Bounds for the Markov constant of $x=[a_0;a_1,a_2,...]$ can also be given by $\sqrt{p^2+4}\le M(x)<p+2$ with $p=\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{a}_n$. This implies that $M(x)=\mathrm{\infty }$ if and only if $(a_k)$ is not bounded and in particular $M(x)<\mathrm{\infty }$ if $x$ is a quadratic irrational number. A further consequence is $M(e)=\mathrm{\infty }$.

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

For rational numbers $r$ it may be defined $M(r)=0$.

Other results

The values $M(e)=\mathrm{\infty }$ and $\mu (e)=2$ imply that the inequality $0<|e-\frac{p}{q}|<\frac{1}{c{q}^2}$ has for all $c\in {ℝ}^{+}$ infinitely many solutions $\frac{p}{q}\in \mathbb{Q}$ while the inequality $0<|e-\frac{p}{q}|<\frac{1}{q^{2+\epsilon }}$ has for all $\epsilon \in {ℝ}^{+}$ 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:

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

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

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

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

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 $\mathrm{deg}L=1$. Consider the following inequality:

$0<|qx-p|\le \max (|p|,|q|{)}^{-\omega }$ with $p,q\in \mathbb{Z}$ and $\omega \in {ℝ}_0^{+}$.

Define $R$ as the set of all $\omega \in {ℝ}_0^{+}$ 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 $\mathrm{deg}P\le n\in {\mathbb{Z}}^{+}$ 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:

$0<|P(z)|\le H(P{)}^{-\omega }$ with $\omega \in {ℝ}_0^{+}$.

Set $R$ to be the set of all $\omega \in {ℝ}_0^{+}$ 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:

$\omega (z)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{\omega }_n(z).$

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 $n\ge N$ 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 $\omega (z)=\frac{1}{2}$.Template:Rp The number e is an S-number with $\omega (e)=1$. The number π is either an S- or T-number.Template:Rp 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 $x_1,...,x_n\in ℝ$ consider the inequality

$0<|c_1{x}_1+...+c_n{x}_n|\le \max (|c_1|,...,|c_n|{)}^{-\nu }$ with $c_1,...,c_n\in \mathbb{Z}$ and $\nu \in {ℝ}_0^{+}$.

Define $R$ as the set of all $\nu \in {ℝ}_0^{+}$ 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 $\mathbb{Q}$ then $\nu (x_1,...,x_n)=0$.

If $1,x_1,...,x_n$ are linearly independent algebraic numbers over $\mathbb{Q}$ then $\nu (1,x_1,...,x_n)\le n$.

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:

$0<|z-\alpha |\le H(\alpha {)}^{-{\omega }^{*}-1}$ with ${\omega }^{*}\in {ℝ}_0^{+}$.

Set $R$ to be the set of all ${\omega }^{*}\in {ℝ}_0^{+}$ 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:

${\omega }^{*}(z)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{\omega }_n^{*}(z)$.

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.Template:Rp

Simultaneous approximation of real numbers

Given a real number $x\in ℝ$, 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

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

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 $x_1,...,x_n\in ℝ$ 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

$0<|{\alpha }_i-\frac{p_i}{q}|<\frac{1}{q^{\mu }},\mathrm{\forall }i\in \{1,...,n\}$

have only at most finitely many solutions $(\frac{p_1}{q},...,\frac{p_n}{q})\in {\mathbb{Q}}^n$ for $\mu >1+\frac{1}{n}$. This result is due to Wolfgang M. Schmidt.

See also

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