Irrationality measure#Irrationality exponent

{{short description|Function that quantifies how near a number is to being rational}}

File:Dedekind cut- square root of two.png.]]

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<\left|x-\frac pq\right|

for a given real number x\in\R and rational numbers \frac pq 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 pq 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 pq\right|

has at most only finitely many solutions \frac pq for all \varepsilon>0.{{cite arXiv |eprint=math/0406300 |first=Jonathan |last=Sondow |title=Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik |year=2004}}

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.{{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 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 pq \approx x with p,q\in\N yields n+1 exact decimal digits, then

:\frac{1}{10^n} \ge \left| x- \frac{p}{q} \right| \ge \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)\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 \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.{{rp|246}}

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)\le\mu, then it follows \mu(x^{1/ 2})\le 2\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:

:\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} \tfrac1{n}{\ln |q_n|} \le \sigma and \lim_{n\to\infty} \tfrac1{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:{{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}}

:\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.

class="wikitable sortable"

|+

! rowspan="2" |Number x

! colspan="2" |Irrationality exponent \mu(x)

! rowspan="2" |Notes

Lower bound

!Upper bound

Rational number p/q with p\in \mathbb{Z}, q\in\mathbb Z^+

| colspan="2" style="text-align: center;" |1

|Every rational number p/q has an irrationality exponent of exactly 1.

Irrational algebraic number \alpha

| colspan="2" style="text-align: center;" |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}^+

| colspan="2" style="text-align: center;" |2

| rowspan="3" |If the elements a_n of the simple continued fraction expansion of an irrational number x are bounded above a_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}^+

| colspan="2" style="text-align: center;" |2

\tanh(1/k), k\in\mathbb{Z}^+

| colspan="2" style="text-align: center;" |2

S(b) with b\geq 2

| colspan="2" style="text-align: center;" |2

|S(b):=\sum_{k=0}^\infty b^{-2^k}with b\in\Z, has continued fraction terms which do not exceed a fixed constant.{{Cite journal |last=Shallit |first=Jeffrey |date=1979-05-01 |title=Simple continued fractions for some irrational numbers |url=https://dx.doi.org/10.1016/0022-314X%2879%2990040-4 |journal=Journal of Number Theory |volume=11 |issue=2 |pages=209–217 |doi=10.1016/0022-314X(79)90040-4 |issn=0022-314X}}{{Cite journal |last=Shallit |first=J. O |date=1982-04-01 |title=Simple continued fractions for some irrational numbers, II |url=https://dx.doi.org/10.1016/0022-314X%2882%2990047-6 |journal=Journal of Number Theory |volume=14 |issue=2 |pages=228–231 |doi=10.1016/0022-314X(82)90047-6 |issn=0022-314X}}

T(b) with b\geq 2{{Cite journal |last=Bugeaud |first=Yann |date=2011 |title=On the rational approximation to the Thue–Morse–Mahler numbers |url=https://aif.centre-mersenne.org/item/AIF_2011__61_5_2065_0/ |journal=Annales de l'Institut Fourier |volume=61 |issue=5 |pages=2065–2076 |doi=10.5802/aif.2666 |issn=1777-5310}}

| colspan="2" style="text-align: center;" |2

|T(b):=\sum_{k=0}^\infty t_kb^{-k} where t_k is the Thue–Morse sequence and b\in\Z. See Prouhet-Thue-Morse constant.

\ln(2){{Cite web |last=Weisstein |first=Eric W. |title=Irrationality Measure |url=https://mathworld.wolfram.com/IrrationalityMeasure.html |access-date=2020-10-14 |website=mathworld.wolfram.com |language=en}}{{Cite journal|last=Nesterenko|first=Yu. V.|date=2010-10-01|title=On the irrationality exponent of the number ln 2|url=https://doi.org/10.1134/S0001434610090257|journal=Mathematical Notes|language=en|volume=88|issue=3|pages=530–543|doi=10.1134/S0001434610090257|s2cid=120685006|issn=1573-8876|url-access=subscription}}

|2

|3.57455...

| rowspan="2" |There are other numbers of the form \ln (a/b) for which bounds on their irrationality exponents are known.{{Cite journal |last=Wu |first=Qiang |date=2003 |title=On the Linear Independence Measure of Logarithms of Rational Numbers |url=https://www.jstor.org/stable/4099938 |journal=Mathematics of Computation |volume=72 |issue=242 |pages=901–911 |doi=10.1090/S0025-5718-02-01442-4 |jstor=4099938 |issn=0025-5718}}{{Cite journal |last1=Bouchelaghem |first1=Abderraouf |last2=He |first2=Yuxin |last3=Li |first3=Yuanhang |last4=Wu |first4=Qiang |date=2024-03-01 |title=On the linear independence measures of logarithms of rational numbers. II |url=https://jkms.kms.or.kr/journal/view.html?uid=2943 |journal=J. Korean Math. Soc. |language=en |volume=61 |issue=2 |pages=293–307 |doi=10.4134/JKMS.j230133}}{{Cite journal |last=Sal’nikova |first=E. S. |date=2008-04-01 |title=Diophantine approximations of log 2 and other logarithms |url=https://link.springer.com/article/10.1134/S0001434608030097 |journal=Mathematical Notes |language=en |volume=83 |issue=3 |pages=389–398 |doi=10.1134/S0001434608030097 |issn=1573-8876|url-access=subscription }}

\ln(3){{Cite web|title=Symmetrized polynomials in a problem of estimating of the irrationality measure of number ln 3|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cheb&paperid=619&option_lang=eng|access-date=2020-10-14|website=www.mathnet.ru}}

|2

|5.11620...

5\ln(3/2){{cite arXiv |last=Polyanskii |first=Alexandr |title=On the irrationality measure of certain numbers |date=2015-01-27 |class=math.NT |eprint=1501.06752}}

|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}{{Cite journal|last=Polyanskii|first=A. A.|date=2018-03-01|title=On the Irrationality Measures of Certain Numbers. II|url=https://doi.org/10.1134/S0001434618030306|journal=Mathematical Notes|language=en|volume=103|issue=3|pages=626–634|doi=10.1134/S0001434618030306|s2cid=125251520|issn=1573-8876|url-access=subscription}}{{Cite journal |last=Androsenko |first=V. A. |date=2015 |title=Irrationality measure of the number \frac{\pi}{\sqrt{3}} |url=https://iopscience.iop.org/article/10.1070/IM2015v079n01ABEH002731 |journal=Izvestiya: Mathematics |language=en |volume=79 |issue=1 |pages=1–17 |doi=10.1070/im2015v079n01abeh002731 |issn=1064-5632 |s2cid=123775303 |via=|url-access=subscription }}

|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{{cite journal|last1=Zeilberger|first1=Doron|last2=Zudilin|first2=Wadim|date=2020-01-07|title=The irrationality measure of π is at most 7.103205334137...|journal=Moscow Journal of Combinatorics and Number Theory|volume=9|issue=4|pages=407–419|doi=10.2140/moscow.2020.9.407|arxiv=1912.06345|s2cid=209370638}}

|2

|7.10320...

|It has been proven that if the Flint Hills series \displaystyle\sum^\infty_{n=1}\frac{\csc^2 n}{n^3} (where n is in radians) converges, then \pi's irrationality exponent is at most 5/2{{cite arXiv |first=Max A. |last=Alekseyev |title=On convergence of the Flint Hills series |eprint=1104.5100 |date=2011 |class=math.CA }}{{MathWorld|FlintHillsSeries|Flint Hills Series}} and that if it diverges, the irrationality exponent is at least 5/2.{{cite arXiv |first=Alex |last=Meiburg| title=Bounds on Irrationality Measures and the Flint-Hills Series|eprint=2208.13356| date=2022 |class=math.NT}}

\pi^2{{Cite journal|last=Zudilin|first=Wadim|date=2014-06-01|title=Two hypergeometric tales and a new irrationality measure of ζ(2)|journal=Annales mathématiques du Québec|volume=38|issue=1|pages=101–117|doi=10.1007/s40316-014-0016-0|arxiv=1310.1526|s2cid=119154009|issn=2195-4763}}

|2

|5.09541...

|\pi^2 and \zeta(2) are linearly dependent over \mathbb{Q}. \left(\zeta(2) = \frac{\pi^2}{6}\right), also see the Basel problem.

\arctan(1/2){{Cite journal |last1=Bashmakova |first1=M. G. |last2=Salikhov |first2=V. Kh. |date=2019 |title=Об оценке меры иррациональности arctg 1/2 |url= |journal=Чебышевский сборник |volume=20 |issue=4 (72) |pages=58–68 |issn=2226-8383}}

|2

|9.27204...

| rowspan="2" |There are many other numbers of the form \arctan(1/k) for which bounds on their irrationality exponents are known.{{Cite web |last=Tomashevskaya |first=E. B. |title=On the irrationality measure of the number log 5+pi/2 and some other numbers |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cheb&paperid=245&option_lang=eng |access-date=2020-10-14 |website=www.mathnet.ru}}{{Cite journal |last1=Salikhov |first1=Vladislav K. |last2=Bashmakova |first2=Mariya G. |date=2022 |title=On rational approximations for some values of arctan(s/r) for natural s and r, s |url=https://projecteuclid.org/journals/moscow-journal-of-combinatorics-and-number-theory/volume-11/issue-2/On-rational-approximations-for-some-values-of-arctans-r-for/10.2140/moscow.2022.11.181.short |journal=Moscow Journal of Combinatorics and Number Theory |volume=11 |issue=2 |pages=181–188 |doi=10.2140/moscow.2022.11.181 |issn=2220-5438|url-access=subscription }}

\arctan(1/3){{Cite journal |last1=Salikhov |first1=V. Kh. |last2=Bashmakova |first2=M. G. |date=2020-12-01 |title=On Irrationality Measure of Some Values of $\operatorname{arctg} \frac{1}{n}$ |url=https://link.springer.com/article/10.3103/S1066369X2012004X |journal=Russian Mathematics |language=en |volume=64 |issue=12 |pages=29–37 |doi=10.3103/S1066369X2012004X |issn=1934-810X|url-access=subscription }}

|2

|5.94202...

Apéry's constant \zeta(3)

|2

|5.51389...

|

\Gamma(1/4){{cite book |last=Waldschmidt |first=Michel |chapter=Elliptic Functions and Transcendence |date=2008 |title=Surveys in Number Theory |pages=143–188 |chapter-url=https://hal.science/hal-00407231 |access-date=2024-09-10 |series=Developments in Mathematics |volume=17 |publisher=Springer Verlag}}

|2

| data-sort-value="100" |10330

|

Cahen's constant C{{Cite journal |last1=Duverney |first1=Daniel |last2=Shiokawa |first2=Iekata |date=2020-01-01 |title=Irrationality exponents of numbers related with Cahen's constant |url=https://link.springer.com/article/10.1007/s00605-019-01335-0 |journal=Monatshefte für Mathematik |language=en |volume=191 |issue=1 |pages=53–76 |doi=10.1007/s00605-019-01335-0 |issn=1436-5081|url-access=subscription }}

| colspan="2" style="text-align: center;" |3

|

Champernowne constants C_b in base b\geq2{{Cite journal|last=Amou|first=Masaaki|date=1991-02-01|title=Approximation to certain transcendental decimal fractions by algebraic numbers|journal=Journal of Number Theory|language=en|volume=37|issue=2|pages=231–241|doi=10.1016/S0022-314X(05)80039-3|issn=0022-314X|doi-access=free}}

| colspan="2" data-sort-value="101" style="text-align: center;" |b

|Examples include C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]

Liouville numbers L

| colspan="2" data-sort-value="102" style="text-align: center;" |\infty

|The Liouville numbers are precisely those numbers having infinite irrationality exponent.{{rp|248}}

Irrationality base

The irrationality base or Sondow irrationality measure is obtained by setting f(q,\beta)=\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 \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,{{Rp|pages=|page=20}} it is known that \beta(e^\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}}

Other irrationality measures

=Markov constant=

{{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\in\R\setminus \mathbb Q it is the factor by which Dirichlet's approximation theorem can be improved for x. Namely if c is a positive real number, then the inequality

:0<\left|x-\frac pq\right|<\frac{1}{cq^2}

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

Dirichlet's approximation theorem implies M(x)\ge1 and Hurwitz's theorem gives M(x)\ge \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(\varphi)=\sqrt 5. It is also M(\sqrt2)=2\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)=\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}\le M(x) with p=\limsup_{n\to\infty}a_n.{{Cite journal |last=Hancl |first=Jaroslav |date=January 2016 |title=Second basic theorem of Hurwitz |journal=Lithuanian Mathematical Journal |volume=56 |pages=72–76 |doi=10.1007/s10986-016-9305-4 |s2cid=124639896}} 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 pq\right|<\frac{1}{cq^2} has for all c\in\R^+ infinitely many solutions \frac pq \in \mathbb Q while the inequality 0<\left|e-\frac pq\right|<\frac{1}{q^{2+\varepsilon}} has for all \varepsilon\in\R^+ only at most finitely many solutions \frac pq \in \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}}

:0<\left|e-\frac pq\right|<\frac{c\ln\ln q}{q^2\ln q}

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

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 pq\in\mathbb Q: (see Khinchin's theorem)

:0<\left|x-\frac pq\right|<\frac{1}{q^2\ln q}

Mahler's generalization

{{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.

=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|\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 \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)=\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\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)=\tfrac12.{{Rp|pages=|page=86}} The number e is an S-number with \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.

=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_1x_1+...+c_nx_n|\le\max(|c_1|,...,|c_n|)^{-\nu} with c_1,...,c_n\in\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.{{cite arXiv |last=Waldschmidt |first=Michel |title=Open Diophantine Problems |date=2004-01-24 |eprint=math/0312440}}

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

= Simultaneous approximation of real numbers =

{{Main|Subspace theorem}}

Given a real number x\in \R, an irrationality measure of x quantifies how well it can be approximated by rational numbers \frac pq 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 pq\right|<\frac {1}{q^\mu}

has only at most finitely many solutions \frac pq\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<\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 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}}

See also

References