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 is a measure of how "closely" it can be approximated by rationals.
If a function , defined for , takes positive real values and is strictly decreasing in both variables, consider the following inequality:
:
for a given real number and rational numbers with . Define as the set of all for which only finitely many exist, such that the inequality is satisfied. Then is called an irrationality measure of with regard to If there is no such and the set is empty, is said to have infinite irrationality measure .
Consequently, the inequality
:
has at most only finitely many solutions for all .{{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 , a definition adapting the one of Liouville numbers — the irrationality exponent is defined for real numbers to be the supremum of the set of such that is satisfied by an infinite number of coprime integer pairs with .{{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 satisfying the above inequality yields good approximations of . Conversely, if , then there are at most finitely many coprime with that satisfy the inequality.
For example, whenever a rational approximation with yields exact decimal digits, then
:
for any , except for at most a finite number of "lucky" pairs .
A number with irrationality exponent 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 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 for real numbers and rational numbers and . If for some we have , then it follows .{{Rp|pages=|page=368}}
For a real number given by its simple continued fraction expansion with convergents it holds:
:
If we have and for some positive real numbers , then we can establish an upper bound for the irrationality exponent of 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}}
:
= 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.
Irrationality base
The irrationality base or Sondow irrationality measure is obtained by setting
Let
:
for all integers
If no such
If a real number
:
=Examples=
Any real number
The number
The numbers
The number
Although it is not known whether or not
Other irrationality measures
=Markov constant=
{{Main|Markov constant}}
Setting
:
has infinitely many solutions
Dirichlet's approximation theorem implies
This is in fact the best general lower bound since the golden ratio gives
Given
:
Bounds for the Markov constant of with Any number with For rational numbers
=Other results=
The values
:
which is satisfied by infinitely many
This makes the number
:
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
Define
=Transcendence measure=
Mahler's irrationality measure can be generalized as follows: Take
Set
Then Mahler's transcendence measure is given by:
:
The transcendental numbers can now be divided into the following three classes:
If for all
If for all
If there exists a smallest positive integer
The number
Almost all numbers are S-numbers. In fact, almost all real numbers give
=Linear independence measure=
Another generalization of Mahler's irrationality measure gives a linear independence measure. For real numbers
Define
If the
If
It is further
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
Set
Then Koksma's transcendence measure is given by:
:
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
:
has only at most finitely many solutions
This can be generalized: Given a set of real numbers
:
have only at most finitely many solutions
See also
References
{{reflist}}{{Number theory}}
{{DEFAULTSORT:Liouville Number}}
Category:Diophantine approximation
Category:Transcendental numbers