Irrationality measure#Irrationality exponent

The irrationality exponent or Liouville–Roth irrationality measure is given by setting $f(q\; ,\backslash mu)=q^\{-\backslash mu\}$, a definition adapting the one of Liouville numbers — the irrationality exponent $\backslash mu(x)$ is defined for real numbers $x$ to be the supremum of the set of $\backslash mu$ such that is satisfied by an infinite number of coprime integer pairs $(p,q)$ with $q>0$.{{Cite book |last1=Parshin |first1=A. N. |url=https://books.google.com/books?id=aWfwCAAAQBAJ |title=Number Theory IV: Transcendental Numbers |last2=Shafarevich |first2=I. R. |date=2013-03-09 |publisher=Springer Science & Business Media |isbn=978-3-662-03644-0 |language=en}}{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=Cambridge University Press | year=2012 | isbn=978-0-521-11169-0 | zbl=1260.11001 | mr=2953186 | doi=10.1017/CBO9781139017732}}{{rp|246}}

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

For example, whenever a rational approximation $\backslash frac\; pq\; \backslash approx\; x$ with $p,q\backslash in\backslash N$ yields $n+1$ exact decimal digits, then

:$\backslash frac\{1\}\{10^n\}\; \backslash ge\; \backslash left|\; x-\; \backslash frac\{p\}\{q\}\; \backslash right|\; \backslash ge\; \backslash frac\{1\}\{q^\{\backslash mu(x)+\backslash varepsilon\}\}$

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

A number $x\backslash in\backslash mathbb\; R$ with irrationality exponent $\backslash mu(x)\backslash le\; 2$ is called a diophantine number,{{Cite web |last=Tao |first=Terence |date=2009 |title=245B, Notes 9: The Baire category theorem and its Banach space consequences |url=https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/ |access-date=2024-09-08 |website=What's new |language=en}} while numbers with $\backslash mu(x)=\backslash infty$ 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.{{rp|246}}

It is $\backslash mu(x)=\backslash mu(rx+s)$ for real numbers $x$ and rational numbers $r\backslash neq\; 0$ and $s$. If for some $x$ we have $\backslash mu(x)\backslash le\backslash mu$, then it follows $\backslash mu(x^\{1/\; 2\})\backslash le\; 2\backslash mu$.{{Rp|pages=|page=368}}

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:

:$\backslash mu(x)=1+\backslash limsup\_\{n\backslash to\backslash infty\}\backslash frac\{\backslash ln\; q\_\{n+1\}\}\{\backslash ln\; q\_n\}=2+\backslash limsup\_\{n\backslash to\backslash infty\}\backslash frac\{\backslash ln\; a\_\{n+1\}\}\{\backslash ln\; q\_n\}\; .$

If we have $\backslash limsup\_\{n\backslash to\backslash infty\}\; \backslash tfrac1\{n\}\{\backslash ln\; |q\_n|\}\; \backslash le\; \backslash sigma$ and $\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash tfrac1\{n\}\{\backslash ln\; |q\_n\; x-p\_n|\}\; =\; -\; \backslash tau$ for some positive real numbers $\backslash sigma,\backslash tau$, then we can establish an upper bound for the irrationality exponent of $x$ by:{{Cite book |last=Chudnovsky |first=G. V. |chapter=Hermite-padé approximations to exponential functions and elementary estimates of the measure of irrationality of π |date=1982 |editor-last=Chudnovsky |editor-first=David V. |editor2-last=Chudnovsky |editor2-first=Gregory V. |title=The Riemann Problem, Complete Integrability and Arithmetic Applications |chapter-url=https://link.springer.com/chapter/10.1007/BFb0093516 |series=Lecture Notes in Mathematics |volume=925 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=299–322 |doi=10.1007/BFb0093516 |isbn=978-3-540-39152-4}}

:$\backslash mu(x)\backslash le\; 1\; +\; \backslash frac\backslash sigma\backslash tau$

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

The irrationality base or Sondow irrationality measure is obtained by setting $f(q,\backslash beta)=\backslash beta^\{-q\}$.{{cite arXiv |last=Sondow |first=Jonathan |title=An irrationality measure for Liouville numbers and conditional measures for Euler's constant |date=2003-07-23 |eprint=math/0307308}} It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding $\backslash beta(x)=1$ for all other real numbers:

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

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

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

If no such $\backslash beta$ exists, then $\backslash beta(x)=\backslash infty$ 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:

:$\backslash beta(x)=\backslash limsup\_\{n\backslash to\backslash infty\}\backslash frac\{\; \backslash ln\; q\_\{n+1\}\}\; \{q\_n\}=\backslash limsup\_\{n\backslash to\backslash infty\}\backslash frac\{\backslash ln\; a\_\{n+1\}\}\{q\_n\}$.

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

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

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

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

Although it is not known whether or not $e^\backslash pi$ is a Liouville number,{{Rp|pages=|page=20}} it is known that $\backslash beta(e^\backslash pi)=1$.{{Cite book |last=Borwein |first=Jonathan M. |url=https://books.google.com/books?id=i8dlAQAACAAJ |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |date=1987 |publisher=Wiley |pages= |language=en}}{{Rp|pages=|page=371}}

{{Main|Markov constant}}

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

:

has infinitely many solutions $\backslash frac\; pq\backslash in\backslash mathbb\; Q$. If $c>M(x)$ there are at most finitely many solutions.

Dirichlet's approximation theorem implies $M(x)\backslash ge1$ and Hurwitz's theorem gives $M(x)\backslash ge\; \backslash sqrt5$ both for irrational $x$.{{cite journal |last=Hurwitz |first=A. |author-link=Adolf Hurwitz |year=1891 |title=Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions) |url=https://gdz.sub.uni-goettingen.de/id/PPN235181684_0039 |journal=Mathematische Annalen |language=de |volume=39 |issue=2 |pages=279–284 |doi=10.1007/BF01206656 |jfm=23.0222.02 |s2cid=119535189|url-access=subscription }}

This is in fact the best general lower bound since the golden ratio gives $M(\backslash varphi)=\backslash sqrt\; 5$. It is also $M(\backslash sqrt2)=2\backslash sqrt\; 2$.

Given $x\; =\; [a\_0;\; a\_1,\; a\_2,\; ...]$ by its simple continued fraction expansion, one may obtain:{{Cite book |last=LeVeque |first=William |title=Fundamentals of Number Theory |publisher=Addison-Wesley Publishing Company, Inc. |year=1977 |isbn=0-201-04287-8 |pages=251–254}}

:$M(x)=\backslash limsup\_\{n\backslash to\backslash infty\}\{([a\_\{n+1\};\; a\_\{n+2\},\; a\_\{n+3\},\; ...]\; +\; [0;\; a\_\{n\},\; a\_\{n-1\},\; ...,a\_2,a\_1])\}.$

Bounds for the Markov constant of $x\; =\; [a\_0;\; a\_1,\; a\_2,\; ...]$ can also be given by

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

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

The values $M(e)=\backslash infty$ and $\backslash mu(e)=2$ imply that the inequality has for all $c\backslash in\backslash R^+$ infinitely many solutions $\backslash frac\; pq\; \backslash in\; \backslash mathbb\; Q$ while the inequality has for all $\backslash varepsilon\backslash in\backslash R^+$ only at most finitely many solutions $\backslash frac\; pq\; \backslash in\; \backslash mathbb\; Q$ . This gives rise to the question what the best upper bound is. The answer is given by:{{Cite journal |last=Davis |first=C. S. |date=1978 |title=Rational approximations to e |url=https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/rational-approximations-to-e/0A59D34AF70DE5ED9A5F3FB1E3703976 |journal=Journal of the Australian Mathematical Society |language=en |volume=25 |issue=4 |pages=497–502 |doi=10.1017/S1446788700021480 |issn=1446-8107}}

:

which is satisfied by infinitely many $\backslash frac\; pq\; \backslash in\; \backslash mathbb\; Q$ for $c>\backslash tfrac12$ but not for .

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

:

{{Main|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.

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

with $p,q\backslash in\backslash mathbb\; Z$ and $\backslash omega\backslash in\backslash R^+\_0$.

Define $R$ as the set of all $\backslash omega\backslash in\backslash R^+\_0$ for which infinitely many solutions $p,q\backslash in\backslash mathbb\; Z$ exist, such that the inequality is satisfied. Then $\backslash omega\_1(x)=\backslash sup\; M$ is Mahler's irrationality measure. It gives $\backslash omega\_1(p/q)=0$ for rational numbers, $\backslash omega\_1(\backslash alpha)=1$ for algebraic irrational numbers and in general $\backslash omega\_1(x)=\backslash mu(x)-1$, where $\backslash mu(x)$ denotes the irrationality exponent.

Mahler's irrationality measure can be generalized as follows: Take $P$ to be a polynomial with $\backslash deg\; P\; \backslash le\; n\backslash in\backslash mathbb\; Z^+$ and integer coefficients $a\_i\backslash in\backslash mathbb\; Z$. Then define a height function $H(P)=\backslash max(|a\_0|,|a\_1|,...,|a\_n|)$ and consider for complex numbers $z$ the inequality:

with $\backslash omega\backslash in\backslash R^+\_0$.

Set $R$ to be the set of all $\backslash omega\backslash in\backslash R^+\_0$ for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define $\backslash omega\_n(z)=\; \backslash sup\; R$ for all $n\backslash in\backslash mathbb\; Z^+$ with $\backslash omega\_1(z)$ being the above irrationality measure, $\backslash omega\_2(z)$ being a non-quadraticity measure, etc.

Then Mahler's transcendence measure is given by:

:$\backslash omega(z)=\backslash limsup\_\{n\backslash to\backslash infty\}\backslash omega\_n(z).$

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

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

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

If there exists a smallest positive integer $N$ such that for all $n\backslash ge\; N$ the $\backslash omega\_n(z)$ 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 $\backslash omega(z)=0$.

Almost all numbers are S-numbers. In fact, almost all real numbers give $\backslash omega(x)=1$ while almost all complex numbers give $\backslash omega(z)=\backslash tfrac12$.{{Rp|pages=|page=86}} The number e is an S-number with $\backslash omega(e)=1$. The number π is either an S- or T-number.{{Rp|pages=|page=86}} The U-numbers are a set of measure 0 but still uncountable.{{Cite book |last1=Burger |first1=Edward B. |url=https://books.google.com/books?id=TRNxKFMuNPkC |title=Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory |last2=Tubbs |first2=Robert |date=2004-07-28 |publisher=Springer Science & Business Media |isbn=978-0-387-21444-3 |language=en}} 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. For real numbers $x\_1,...,x\_n\backslash in\; \backslash R$ consider the inequality

with $c\_1,...,c\_n\backslash in\backslash Z$ and $\backslash nu\backslash in\backslash R^+\_0$.

Define $R$ as the set of all $\backslash nu\backslash in\backslash R^+\_0$ for which infinitely many solutions $c\_1,...c\_n\; \backslash in\backslash mathbb\; Z$ exist, such that the inequality is satisfied. Then $\backslash nu(x\_1,...,x\_n)=\; \backslash sup\; R$ is the linear independence measure.

If the $x\_1,...,x\_n$ are linearly dependent over $\backslash mathbb\backslash Q$ then $\backslash nu(x\_1,...,x\_n)=0$.

If $1,x\_1,...,x\_n$ are linearly independent algebraic numbers over $\backslash mathbb\backslash Q$ then $\backslash nu(1,x\_1,...,x\_n)\backslash le\; n$.{{cite arXiv |last=Waldschmidt |first=Michel |title=Open Diophantine Problems |date=2004-01-24 |eprint=math/0312440}}

It is further $\backslash nu(1,x)=\backslash omega\_1(x)=\backslash mu(x)-1$.

Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of complex numbers by algebraic numbers.{{Cite book |last=Baker |first=Alan |title=Transcendental number theory |date=1979 |publisher=Cambridge Univ. Pr |isbn=978-0-521-20461-3 |edition=Repr. with additional material |location=Cambridge}}

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

with $\backslash omega^*\backslash in\backslash R^+\_0$.

Set $R$ to be the set of all $\backslash omega^*\backslash in\backslash R^+\_0$ for which infinitely many such algebraic numbers $\backslash alpha$ exist, that keep the inequality satisfied. Further define $\backslash omega\_n^*(z)=\backslash sup\; R$ for all $n\backslash in\backslash mathbb\; Z^+$ with $\backslash omega\_1^*(z)$ being an irrationality measure, $\backslash omega\_2^*(z)$ being a non-quadraticity measure, etc.

Then Koksma's transcendence measure is given by:

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

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.{{Rp|pages=|page=87}}

{{Main|Subspace theorem}}

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

:

has only at most finitely many solutions $\backslash frac\; pq\backslash in\; \backslash mathbb\; Q$ for $\backslash mu>2$. This is known as Roth's theorem.

This can be generalized: Given a set of real numbers $x\_1,...,x\_n\backslash in\; \backslash R$ one can quantify how well they can be approximated simultaneously by rational numbers $\backslash frac\{p\_1\}\{q\},...,\backslash frac\{p\_n\}\{q\}$ with the same denominator $q\backslash in\backslash mathbb\; Z^+$. If the $x\_i=\backslash alpha\_i$ are taken to be algebraic numbers, such that $1,\backslash alpha\_1,...,\backslash alpha\_n$ are linearly independent over the rational numbers $\backslash mathbb\; Q$ it follows that the inequalities

:

have only at most finitely many solutions $\backslash left(\backslash frac\{p\_1\}\{q\},...,\backslash frac\{p\_n\}\{q\}\backslash right)\backslash in\; \backslash mathbb\; Q^n$ for $\backslash mu>\; 1\; +\; \backslash frac\; 1n$. This result is due to Wolfgang M. Schmidt.{{Cite journal |last=Schmidt |first=Wolfgang M. |date=1972 |title=Norm Form Equations |url=https://www.jstor.org/stable/1970824 |journal=Annals of Mathematics |volume=96 |issue=3 |pages=526–551 |doi=10.2307/1970824 |jstor=1970824 |issn=0003-486X|url-access=subscription }}{{Cite book |last=Schmidt |first=Wolfgang M. |title=Diophantine approximation |date=1996 |publisher=Springer |isbn=978-3-540-09762-4 |series=Lecture notes in mathematics |location=Berlin ; New York}}

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

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Category:Diophantine approximation

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Category:Irrational numbers