Irrationality measure

In mathematics, an irrationality measure of a real number $x$ is a measure of how "closely" it can be approximated by rationals.

Rational approximations to the Square root of 2.

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 $q>0$.

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.

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 $\mu (x^{1/2})\leq 2\mu$.

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 $\mu (x)$		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 $\varphi$.
$e^{2/k},k\in \mathbb {Z} ^{+}$	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 $b\geq 2$	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		$T(b):=\sum _{k=0}^{\infty }t_{k}b^{-k}$ 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...
$5\ln(3/2)$	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 $\mathbb {Q}$.
$\arctan(1/2)$	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 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, it is known that $\beta (e^{\pi })=1$.

Other irrationality measures

Markov constant

Main article: 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

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.

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 $\omega (z)={\tfrac {1}{2}}$. The number e is an S-number with $\omega (e)=1$. The number π is either an S- or T-number. 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.

Simultaneous approximation of real numbers

Main article: Subspace theorem

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.

See also

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