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Irrationality Measure


Let x be a real number, and let R be the set of positive real numbers mu for which

 0<|x-p/q|<1/(q^mu)
(1)

has (at most) finitely many solutions p/q for p and q integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and x is no longer approximable by rational numbers,

 mu(x)=inf_(mu in R)mu,
(2)

where inf_(mu in R)mu is the infimum. If the set R is empty, then mu(x) is defined to be mu(x)=infty, and x is called a Liouville number. There are three possible regimes for nonempty R:

 {mu(x)=1   if x is rational; mu(x)=2   if x is algebraic of degree >1; mu(x)>=2   if x is transcendental,
(3)

where the transitional case mu(x)=2 can correspond to x being either algebraic of degree >1 or x being transcendental. Showing that mu(x)=2 for x an algebraic number is a difficult result for which Roth was awarded the Fields medal.

The definition of irrationality measure is equivalent to the statement that if x has irrationality measure mu, then mu is the smallest number such that the inequality

 |x-p/q|>1/(q^(mu+epsilon))
(4)

holds for any epsilon>0 and all integers p and q with q sufficiently large.

The irrationality measure of an irrational number x can be given in terms of its simple continued fraction expansion x=[a_0,a_1,a_2,...] and its convergents p_n/q_n as

mu(x)=1+lim sup_(n->infty)(lnq_(n+1))/(lnq_n)
(5)
=2+lim sup_(n->infty)(lna_(n+1))/(lnq_n)
(6)

(Sondow 2004). For example, the golden ratio phi has

 mu(phi)=2,
(7)

which follows immediately from (6) and the simple continued fraction expansion phi=[1,1,1,...].

Exact values include mu(L)=infty for L Liouville's constant and mu(e)=2 (Borwein and Borwein 1987, pp. 364-365). The best known upper bounds for other common constants as of mid-2020 are summarized in the following table, where zeta(3) is Apéry's constant, Ln_q(2) and h_q(1) are q-harmonic series, and the lower bounds are 2.

constant xupper boundreference
pi7.10320534Zeilberger and Zudilin (2020)
pi^25.09541179Zudilin (2013)
ln23.57455391Marcovecchio (2009)
ln35.116201Bondareva et al. (2018)
zeta(3)5.513891Rhin and Viola (2001)
Ln_q(2)2.9384Matala-Aho et al. (2006)
h_q(1)2.4650Zudilin (2004)

The bound for pi is due to Zeilberger and Zudilin (2020) and improves on the value 7.606308 previously found by Salikhov (2008). It has exact value given as follows. Let N_+/- be the complex conjugate roots of

 108N^3-2359989N^2+138304N-2048=0,
(8)

let N_3 be the positive real root, and let

a_1=ln|N_+/-|-5/2ln2+4-pi/(2sqrt(3))+ln((3sqrt(3))/4)
(9)
=-1.90291648...
(10)
a_3=lnN_3-5/2ln2+4-pi/(2sqrt(3))+ln((3sqrt(3))/4)
(11)
=11.61389004...,
(12)

then the bound is given by

 mu(pi)<=1-(a_3)/(a_1).
(13)

Alekseyev (2011) has shown that the question of the convergence of the Flint Hills series is related to the irrationality measure of pi, and in particular, convergence would imply mu(pi)<=2.5, which is much stronger than the best currently known upper bound.


See also

Algebraic Number, Liouville's Approximation Theorem, Rational Number, Roth's Theorem, Transcendence Degree, Transcendental Number

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References

Alekseyev, M. A. "On Convergence of the Flint Hills Series." http://arxiv.org/abs/1104.5100/. 27 Apr 2011.Amdeberhan, T. and Zeilberger, D. "q-Apéry Irrationality Proofs by q-WZ Pairs." Adv. Appl. Math. 20, 275-283, 1998.Beukers, F. "A Rational Approach to Pi." Nieuw Arch. Wisk. 5, 372-379, 2000.Bondareva, I. V.; Luchin, M. Y.; and Salikhov, V. K. "Symmetrized Polynomials in a Problem of Estimating the Irrationality Measure of the Number ln3." Chebyshevskiĭ Sb. 19, 15-25, 2018.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 3-4, 2004.Borwein, J. M. and Borwein, P. B. "Irrationality Measures." §11.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 362-386, 1987.Finch, S. R. "Liouville-Roth Constants." §2.22 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 171-174, 2003.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press, 1979.Hata, M. "Legendre Type Polynomials and Irrationality Measures." J. reine angew. Math. 407, 99-125, 1990.Hata, M. "Improvement in the Irrationality Measures of pi and pi^2." Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-286, 1992.Hata, M. "Rational Approximations to pi and Some Other Numbers." Acta Arith. 63, 335-349, 1993.Hata, M. "A Note on Beuker's Integral." J. Austral. Math. Soc. 58, 143-153, 1995.Hata, M. "A New Irrationality Measure for zeta(3)." Acta Arith. 92, 47-57, 2000.Marcovecchio, R. "The Rhin-Viola Method for log2." Acta Arith. 139, 147-184, 2009.Matala-Aho, T.; Väänänen, K.; and Zudilin, W. "New Irrationality Measures for q-Logarithms." Math. Comput. 75, 879-889, 2006.Rhin, G. and Viola, C. "On a Permutation Group Related to zeta(2)." Acta Arith. 77, 23-56, 1996.Rhin, G. and Viola, C. "The Group Structure for zeta(3)." Acta Arith. 97, 269-293, 2001.Rukhadze, E. A. "A Lower Bound for the Rational Approximation of ln2 by Rational Numbers." [In Russian]. Vestnik Moskov Univ. Ser. I Math. Mekh., No. 6, 25-29 and 97, 1987.Salikhov, V. Kh. "On the Irrationality Measure of ln3."Dokl. Akad. Nauk 417, 753-755, 2007. Translation in Dokl. Math. 76, No. 3, 955-957, 2007.Salikhov, V. Kh. "On the Irrationality Measure of pi." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008.Sondow, J. "Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik." Proceedings of Journées Arithmétiques, Graz 2003 in the Journal du Theorie des Nombres Bordeaux. http://arxiv.org/abs/math.NT/0406300.Stark, H. M. An Introduction to Number Theory. Cambridge, MA: MIT Press, 1994.van Assche, W. "Little q-Legendre Polynomials and Irrationality of Certain Lambert Series." Jan. 23, 2001. http://wis.kuleuven.be/analyse/walter/qLegend.pdf.Zeilberger, D. and Zudilin, W. "The Irrationality Measure of pi is at Most 7.103205334137...." 8 Jan 2020. https://arxiv.org/abs/1912.06345.Zudilin, V. V. "An Essay on the Irrationality Measures of pi and Other Logarithms." Chebyshevskiĭ Sb. 5, 49-65, 2004.Zudilin, V. V. "On the Irrationality Measure of pi^2." Russian Math. Surveys 68, 1133-1135, 2013.

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Irrationality Measure

Cite this as:

Weisstein, Eric W. "Irrationality Measure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrrationalityMeasure.html

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