Irrationality measure

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Rational approximations to the Square root of 2. Dedekind cut- square root of two.png
Rational approximations to the Square root of 2.

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

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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 . [1]

Irrationality exponent

The irrationality exponent or Liouville–Roth irrationality measure is given by setting , [1] 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 . [2] [3] :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, [4] 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. [3] :246

It is for real numbers and rational numbers and . If for some we have , then it follows . [5] :368

For a real number given by its simple continued fraction expansion with convergents it holds: [1]

If we have and for some positive real numbers , then we can establish an upper bound for the irrationality exponent of by: [6] [7]

Known bounds

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 Irrationality exponent Notes
Lower boundUpper bound
Rational number with 1Every rational number has an irrationality exponent of exactly 1.
Irrational algebraic number 2By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots and the golden ratio .
2If the elements of the simple continued fraction expansion of an irrational number are bounded above by an arbitrary polynomial , then its irrationality exponent is .

Examples include numbers which continued fractions behave predictably such as

and .

2
2
with 2with , has continued fraction terms which do not exceed a fixed constant. [8] [9]
with [10] 2 where is the Thue–Morse sequence and . See Prouhet-Thue-Morse constant.
[11] [12] 23.57455...There are other numbers of the form for which bounds on their irrationality exponents are known. [13] [14] [15]
[11] [16] 25.11620...
[17] 23.43506...There are many other numbers of the form for which bounds on their irrationality exponents are known. [17] This is the case for .
[18] [19] 24.60105...There are many other numbers of the form for which bounds on their irrationality exponents are known. [18] This is the case for .
[11] [20] 27.10320...It has been proven that if the Flint Hills series (where n is in radians) converges, then 's irrationality exponent is at most [21] [22] and that if it diverges, the irrationality exponent is at least . [23]
[11] [24] 25.09541... and are linearly dependent over .
[25] 29.27204...There are many other numbers of the form for which bounds on their irrationality exponents are known. [26] [27]
[28] 25.94202...
Apéry's constant [11] 25.51389...
[29] 210330
Cahen's constant [30] 3
Champernowne constants in base [31] Examples include
Liouville numbers The Liouville numbers are precisely those numbers having infinite irrationality exponent. [3] :248

Irrationality base

The irrationality base or Sondow irrationality measure is obtained by setting . [1] [6] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding for all other real numbers:

Let be an irrational number. If there exist real numbers with the property that for any , there is a positive integer such that

for all integers with then the least such is called the irrationality base of and is represented as .

If no such exists, then and is called a super Liouville number.

If a real number is given by its simple continued fraction expansion with convergents then it holds:

. [1]

Examples

Any real number with finite irrationality exponent has irrationality base , while any number with irrationality base has irrationality exponent and is a Liouville number.

The number has irrationality exponent and irrationality base .

The numbers ( represents tetration, ) have irrationality base .

The number has irrationality base , hence it is a super Liouville number.

Although it is not known whether or not is a Liouville number, [32] :20 it is known that . [5] :371

Other irrationality measures

Markov constant

Setting gives a stronger irrationality measure: the Markov constant . For an irrational number it is the factor by which Dirichlet's approximation theorem can be improved for . Namely if is a positive real number, then the inequality

has infinitely many solutions . If there are at most finitely many solutions.

Dirichlet's approximation theorem implies and Hurwitz's theorem gives both for irrational . [33]

This is in fact the best general lower bound since the golden ratio gives . It is also .

Given by its simple continued fraction expansion, one may obtain: [34]

Bounds for the Markov constant of can also be given by with . [35] This implies that if and only if is not bounded and in particular if is a quadratic irrational number. A further consequence is .

Any number with or has an unbounded simple continued fraction and hence .

For rational numbers it may be defined .

Other results

The values and imply that the inequality has for all infinitely many solutions while the inequality has for all only at most finitely many solutions . This gives rise to the question what the best upper bound is. The answer is given by: [36]

which is satisfied by infinitely many for but not for .

This makes the number alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers the inequality below has infinitely many solutions : [5] (see Khinchin's theorem)

Mahler's generalization

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]

Mahler's irrationality measure

Instead of taking for a given real number the difference with , one may instead focus on term with and with . Consider the following inequality:

with and .

Define as the set of all for which infinitely many solutions exist, such that the inequality is satisfied. Then is Mahler's irrationality measure. It gives for rational numbers, for algebraic irrational numbers and in general , where denotes the irrationality exponent.

Transcendence measure

Mahler's irrationality measure can be generalized as follows: [2] [3] Take to be a polynomial with and integer coefficients . Then define a height function and consider for complex numbers the inequality:

with .

Set to be the set of all for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define for all with being the above irrationality measure, being a non-quadraticity measure, etc.

Then Mahler's transcendence measure is given by:

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

If for all the value of is finite and is finite as well, is called an S-number (of type ).

If for all the value of is finite but is infinite, is called an T-number.

If there exists a smallest positive integer such that for all the are infinite, is called an U-number (of degree ).

The number is algebraic (and called an A-number) if and only if .

Almost all numbers are S-numbers. In fact, almost all real numbers give while almost all complex numbers give . [37] :86 The number e is an S-number with . 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.

Linear independence measure

Another generalization of Mahler's irrationality measure gives a linear independence measure. [2] [13] For real numbers consider the inequality

with and .

Define as the set of all for which infinitely many solutions exist, such that the inequality is satisfied. Then is the linear independence measure.

If the are linearly dependent over then .

If are linearly independent algebraic numbers over then . [32]

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. [3] [37]

For a given complex number consider algebraic numbers of degree at most . Define a height function , where is the characteristic polynomial of and consider the inequality:

with .

Set to be the set of all for which infinitely many such algebraic numbers exist, that keep the inequality satisfied. Further define for all with being an irrationality measure, being a non-quadraticity measure, [17] etc.

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. [37] :87

Simultaneous approximation of real numbers

Given a real number , an irrationality measure of quantifies how well it can be approximated by rational numbers with denominator . If is taken to be an algebraic number that is also irrational one may obtain that the inequality

has only at most finitely many solutions for . This is known as Roth's theorem.

This can be generalized: Given a set of real numbers one can quantify how well they can be approximated simultaneously by rational numbers with the same denominator . If the are taken to be algebraic numbers, such that are linearly independent over the rational numbers it follows that the inequalities

have only at most finitely many solutions for . This result is due to Wolfgang M. Schmidt. [39] [40]

See also

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References

  1. 1 2 3 4 5 Sondow, Jonathan (2004). "Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik". arXiv: math/0406300 .
  2. 1 2 3 Parshin, A. N.; Shafarevich, I. R. (2013-03-09). Number Theory IV: Transcendental Numbers. Springer Science & Business Media. ISBN   978-3-662-03644-0.
  3. 1 2 3 4 5 6 Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139017732. ISBN   978-0-521-11169-0. MR   2953186. Zbl   1260.11001.
  4. Tao, Terence (2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's new. Retrieved 2024-09-08.
  5. 1 2 3 4 Borwein, Jonathan M. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley.
  6. 1 2 Sondow, Jonathan (2003-07-23). "An irrationality measure for Liouville numbers and conditional measures for Euler's constant". arXiv: math/0307308 .
  7. Chudnovsky, G. V. (1982). Chudnovsky, David V.; Chudnovsky, Gregory V. (eds.). "Hermite-padé approximations to exponential functions and elementary estimates of the measure of irrationality of π". The Riemann Problem, Complete Integrability and Arithmetic Applications. Berlin, Heidelberg: Springer: 299–322. doi:10.1007/BFb0093516. ISBN   978-3-540-39152-4.
  8. Shallit, Jeffrey (1979-05-01). "Simple continued fractions for some irrational numbers". Journal of Number Theory. 11 (2): 209–217. doi:10.1016/0022-314X(79)90040-4. ISSN   0022-314X.
  9. Shallit, J. O (1982-04-01). "Simple continued fractions for some irrational numbers, II". Journal of Number Theory. 14 (2): 228–231. doi:10.1016/0022-314X(82)90047-6. ISSN   0022-314X.
  10. Bugeaud, Yann (2011). "On the rational approximation to the Thue–Morse–Mahler numbers". Annales de l'Institut Fourier. 61 (5): 2065–2076. doi:10.5802/aif.2666. ISSN   1777-5310.
  11. 1 2 3 4 5 Weisstein, Eric W. "Irrationality Measure". mathworld.wolfram.com. Retrieved 2020-10-14.
  12. Nesterenko, Yu. V. (2010-10-01). "On the irrationality exponent of the number ln 2". Mathematical Notes. 88 (3): 530–543. doi:10.1134/S0001434610090257. ISSN   1573-8876. S2CID   120685006.
  13. 1 2 Wu, Qiang (2003). "On the Linear Independence Measure of Logarithms of Rational Numbers". Mathematics of Computation. 72 (242): 901–911. doi:10.1090/S0025-5718-02-01442-4. ISSN   0025-5718. JSTOR   4099938.
  14. Bouchelaghem, Abderraouf; He, Yuxin; Li, Yuanhang; Wu, Qiang (2024-03-01). "On the linear independence measures of logarithms of rational numbers. II". J. Korean Math. Soc. 61 (2): 293–307. doi:10.4134/JKMS.j230133.
  15. Sal’nikova, E. S. (2008-04-01). "Diophantine approximations of log 2 and other logarithms". Mathematical Notes. 83 (3): 389–398. doi:10.1134/S0001434608030097. ISSN   1573-8876.
  16. "Symmetrized polynomials in a problem of estimating of the irrationality measure of number ln 3". www.mathnet.ru. Retrieved 2020-10-14.
  17. 1 2 3 Polyanskii, Alexandr (2015-01-27). "On the irrationality measure of certain numbers". arXiv: 1501.06752 [math.NT].
  18. 1 2 Polyanskii, A. A. (2018-03-01). "On the Irrationality Measures of Certain Numbers. II". Mathematical Notes. 103 (3): 626–634. doi:10.1134/S0001434618030306. ISSN   1573-8876. S2CID   125251520.
  19. Androsenko, V. A. (2015). "Irrationality measure of the number \frac{\pi}{\sqrt{3}}". Izvestiya: Mathematics. 79 (1): 1–17. doi:10.1070/im2015v079n01abeh002731. ISSN   1064-5632. S2CID   123775303.
  20. Zeilberger, Doron; Zudilin, Wadim (2020-01-07). "The irrationality measure of π is at most 7.103205334137...". Moscow Journal of Combinatorics and Number Theory. 9 (4): 407–419. arXiv: 1912.06345 . doi:10.2140/moscow.2020.9.407. S2CID   209370638.
  21. Alekseyev, Max A. (2011). "On convergence of the Flint Hills series". arXiv: 1104.5100 [math.CA].
  22. Weisstein, Eric W. "Flint Hills Series". MathWorld .
  23. Meiburg, Alex (2022). "Bounds on Irrationality Measures and the Flint-Hills Series". arXiv: 2208.13356 [math.NT].
  24. Zudilin, Wadim (2014-06-01). "Two hypergeometric tales and a new irrationality measure of ζ(2)". Annales mathématiques du Québec. 38 (1): 101–117. arXiv: 1310.1526 . doi:10.1007/s40316-014-0016-0. ISSN   2195-4763. S2CID   119154009.
  25. Bashmakova, M. G.; Salikhov, V. Kh. (2019). "Об оценке меры иррациональности arctg 1/2". Чебышевский сборник. 20 (4 (72)): 58–68. ISSN   2226-8383.
  26. Tomashevskaya, E. B. "On the irrationality measure of the number log 5+pi/2 and some other numbers". www.mathnet.ru. Retrieved 2020-10-14.
  27. Salikhov, Vladislav K.; Bashmakova, Mariya G. (2022). "On rational approximations for some values of arctan(s/r) for natural s and r, s". Moscow Journal of Combinatorics and Number Theory. 11 (2): 181–188. doi:10.2140/moscow.2022.11.181. ISSN   2220-5438.
  28. Salikhov, V. Kh.; Bashmakova, M. G. (2020-12-01). "On Irrationality Measure of Some Values of $\operatorname{arctg} \frac{1}{n}$". Russian Mathematics. 64 (12): 29–37. doi:10.3103/S1066369X2012004X. ISSN   1934-810X.
  29. Waldschmidt, Michel (2008). "Elliptic Functions and Transcendence". Surveys in Number Theory. Developments in Mathematics. Vol. 17. Springer Verlag. pp. 143–188. Retrieved 2024-09-10.
  30. Duverney, Daniel; Shiokawa, Iekata (2020-01-01). "Irrationality exponents of numbers related with Cahen's constant". Monatshefte für Mathematik. 191 (1): 53–76. doi:10.1007/s00605-019-01335-0. ISSN   1436-5081.
  31. Amou, Masaaki (1991-02-01). "Approximation to certain transcendental decimal fractions by algebraic numbers". Journal of Number Theory. 37 (2): 231–241. doi: 10.1016/S0022-314X(05)80039-3 . ISSN   0022-314X.
  32. 1 2 Waldschmidt, Michel (2004-01-24). "Open Diophantine Problems". arXiv: math/0312440 .
  33. Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions)". Mathematische Annalen (in German). 39 (2): 279–284. doi:10.1007/BF01206656. JFM   23.0222.02. S2CID   119535189.
  34. LeVeque, William (1977). Fundamentals of Number Theory. Addison-Wesley Publishing Company, Inc. pp. 251–254. ISBN   0-201-04287-8.
  35. Hancl, Jaroslav (January 2016). "Second basic theorem of Hurwitz". Lithuanian Mathematical Journal. 56: 72–76. doi:10.1007/s10986-016-9305-4. S2CID   124639896.
  36. Davis, C. S. (1978). "Rational approximations to e". Journal of the Australian Mathematical Society. 25 (4): 497–502. doi:10.1017/S1446788700021480. ISSN   1446-8107.
  37. 1 2 3 4 Baker, Alan (1979). Transcendental number theory (Repr. with additional material ed.). Cambridge: Cambridge Univ. Pr. ISBN   978-0-521-20461-3.
  38. Burger, Edward B.; Tubbs, Robert (2004-07-28). Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory. Springer Science & Business Media. ISBN   978-0-387-21444-3.
  39. Schmidt, Wolfgang M. (1972). "Norm Form Equations". Annals of Mathematics. 96 (3): 526–551. doi:10.2307/1970824. ISSN   0003-486X. JSTOR   1970824.
  40. Schmidt, Wolfgang M. (1996). Diophantine approximation. Lecture notes in mathematics. Berlin ; New York: Springer. ISBN   978-3-540-09762-4.