Irrationality measure

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.

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]

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

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 ${\displaystyle \mathbb {Q} }$ .
${\displaystyle \arctan(1/2)}$ [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

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]

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

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 ${\displaystyle M(r)=0}$ .

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)

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

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]

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.

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.

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

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

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]

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