Irrationality measure

An irrationality measure of a real number $\alpha$ is a measure of how "closely" it can be approximated by rationals. If a function $f(x,\lambda )$, defined for positive real numbers, strictly decreasing in both $x$ and $\lambda$ is given, consider the following inequality:

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

for a given real number $\alpha \in \mathbb {R}$ and rational numbers ${\frac {p}{q}}$ with $p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}$. Define $M$ 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 $\lambda (\alpha )=\inf M$ is called an irrationality measure of $\alpha$ with regard to $f.$ If there is no such $\lambda$ and the set $M$ is empty, $\alpha$ is said to have infinite irrationality measure $\lambda (\alpha )=\infty$.

Consequently the inequality

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

has at most only finitely many solutions ${\frac {p}{q}}$.

Irrationality exponent

The irrationality exponent oder Liouville–Roth irrationality measure is given by setting $f(x,\mu )=x^{-\mu }$, a definition adapting the one of Liouville numbers — the irrationality exponent $\mu (x)$ is defined to be the supremum of the set of $\mu$ such that $0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}$ is satisfied by an infinite number of coprime integer pairs $(p,q)$ with $q>0$. For any value $n\leq \mu (\alpha )$, the infinite set of all rationals $p/q$ satisfying the above inequality yields good approximations of $\alpha$. 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 $\alpha \approx {\frac {p}{q}}$, $p,q\in \mathbb {N}$ 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 $(p,q)$.

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 numbers, have an irrationality exponent equal to 2.

A number $\alpha \in \mathbb {R}$ with irrationality exponent $\mu (\alpha )=2$ is called a diophantine number, while numbers with $\mu (\alpha )=\infty$ are called Liouville numbers.

It is $\mu (\alpha )=\mu (r\alpha +s)$ for real numbers $\alpha$ and rational numbers $r\neq 0$ and $s$.

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

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

Below is a table of known upper and lower bounds for the irrationality exponents of certain numbers.

Number $x$	Irrationality exponent $\mu (\alpha )$		Notes
	Lower bound	Upper bound
Rational number ${\frac {p}{q}}$ with $p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}$	1		Every rational number ${\frac {p}{q}}$ has an irrationality exponent of exactly 1.
Irrational algebraic number $a$	2		By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots like ${\sqrt {2}}$ and the golden ratio $\varphi$.
$e^{2/k},k\in \mathbb {Z} ^{+}$	2		If the elements $a_{n}$ of the simple continued fraction expansion of an irrational number $\alpha$ are bounded above $a_{n}<P(n)$ by an arbitrary polynomial $P$, then its irrationality exponent is $\mu (\alpha )=2$. Examples include numbers which continued fractions behave predictably such as: $e=[2;1,2,1,1,4,1,1,6,1,1,...]$ and $I_{0}(1)/I_{1}(1)=[2;4,6,8,10,12,14,16,18,20,22...]$
$\tanh \left({\frac {1}{k}}\right),k\in \mathbb {Z} ^{+}$	2
$\tan \left({\frac {1}{k}}\right),k\in \mathbb {Z} ^{+}$	2
$T(b)$ with $b\geq 2$	2		$T(b):=\sum _{k=0}^{\infty }t_{k}b^{-k}$ where $t_{k}$ is the $k$-th term of 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 $\ln a$ 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 $\displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}$ (where n is in radians) converges, then $\pi$'s irrationality exponent is at most 2.5; and that if it diverges, the irrationality exponent is at least 2.5.
$\pi ^{2}$ and $\zeta (2)$	2	5.09541...	$\pi ^{2}$ and $\zeta (2)$ are linearly dependent over $\mathbb {Q}$.
$\arctan(1/2)$	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...
$\Gamma (1/4)$	2	10330
Cahen's constant $C$	3
Champernowne constants $C_{b}$ in base $b\geq 2$	$b$		Examples include $C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]$
Liouville numbers $L$	$\infty$		The Liouville numbers are precisely those numbers having infinite irrationality exponent.

Irrationality base

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

Let $\alpha$ 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

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

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

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

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

$\beta (\alpha )=\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{q_{n}}}=\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{q_{n}}}$.

Examples:

Any real number $\alpha$ with finite irrationality exponent $\mu (\alpha )<\infty$ has irrationality base $\beta (\alpha )=1$, while any number with irrationality base $\beta (\alpha )>1$ has irrationality exponent $\mu (\alpha )=\infty$ 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.

Other irrationality measures

Markov constant

Main article: Markov constant

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

$0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{Aq^{2}}}$

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

Dirichlet's approximation theorem implies $M(\alpha )\geq 1$ and Hurwitz's theorem gives $M(\alpha )\geq {\sqrt {5}}$ both for irrational $\alpha .$

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 $\alpha =[a_{0};a_{1},a_{2},...]$ by its simple continued fraction expansion, one may obtain $M(\alpha )=\limsup _{n\to \infty }{([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_{n},a_{n-1},...,a_{2},a_{1}])}$.

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

Any number with $\mu (\alpha )>2$ oder $\beta (\alpha )>1$ has an unbounded simple continued fraction and hence $M(\alpha )=\infty$.

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 $0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{Aq^{2}}}$ has for all $A\in \mathbb {R} ^{+}$ 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:

$0<\left|e-{\frac {p}{q}}\right|<{\frac {\ln \ln q}{(2+\varepsilon )q^{2}\ln q}}$

which is satisfied by infinitely many ${\frac {p}{q}}\in \mathbb {Q}$ for $\varepsilon <0$ but not for $\varepsilon >0$.

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

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

Mahler's generalization

Main article: Transcendental number theory

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 $\alpha$ the difference $|\alpha -p/q|$ with $p/q\in \mathbb {Q}$, one may instead focus on term $|q\alpha -p|=|L(\alpha )|$ with $p,q\in \mathbb {Z}$ and $\deg L=1$. Consider the following inequality:

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

Define $M$ 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 $\omega _{1}(\alpha )=\sup M$ is Mahler's irrationality measure. It gives $\omega _{1}(p/q)=0$ for rational numbers, $\omega _{1}(x_{0})=1$ for algebraic irrational numbers and in general $\omega _{1}(\alpha )=\mu (\alpha )-1$, where $\mu (\alpha )$ 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 real numbers $\alpha$ the inequality:

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

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

Then Mahler's transcendence measure is given by:

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

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

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

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

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

The number $\alpha$ is algebraic if and only if $\omega (\alpha )=0$.

Almost all numbers are S-numbers, however the Liouville numbers are a subset of the U-numbers.

Linear independence measure

Another generalization of Mahler's irrationality measure gives a linear independence measure. For real numbers $\alpha _{1},...,\alpha _{n}\in \mathbb {R}$ consider the inequality

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

Define $M$ 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 $\nu (\alpha _{1},...,\alpha _{n})=\sup M$ is the linear independence measure.

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

If $1,\alpha _{1},...,\alpha _{n}$ are algebraic and linearly independent over $\mathbb {\mathbb {Q} }$ then $\nu (1,\alpha _{1},...,\alpha _{n})\leq n$.

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

Other generalizations

Koksma’s generalization

Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of real numbers by algebraic numbers.

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

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

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

Then Koksma's transcendence measure is given by:

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

Simultaneous approximations of real numbers

Main article: Subspace theorem

Given a real number $\alpha \in \mathbb {R}$ an irrationality measure of $\alpha$ quantifies how well it can be approximated by rational numbers ${\frac {p}{q}}$ with denominator $q\in \mathbb {Z} ^{+}$. If $\alpha$ 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 $\alpha _{1},...,\alpha _{n}\in \mathbb {R}$ 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 $\alpha _{i}$ are taken to be an algebraic number that, 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.

See also

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