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Let 
be a real number, and let 
be the set of positive real numbers 
for which
 | (1) |
has (at most) finitely many solutions 
for 
and 
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 
is no longer approximable by rational numbers,
 | (2) |
where 
is the infimum. If the set 
is empty, then 
is defined to be 
, and 
is called a Liouville number. There are three possible regimes for nonempty 
:
 | (3) |
where the transitional case 
can correspond to 
being either algebraic of degree 
or 
being transcendental. Showing that 
for 
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 
has irrationality measure 
, then 
is the smallest number such that the inequality
 | (4) |
holds for any 
and all integers 
and 
with 
sufficiently large.
The irrationality measure of an irrational number 
can be given in terms of its simple continued fraction expansion 
and its convergents 
as
 |  |  | (5) |
 |  |  | (6) |
(Sondow 2004). For example, the golden ratio 
has
 | (7) |
which follows immediately from (6) and the simple continued fraction expansion 
.
Exact values include 
for 
Liouville's constant and 
(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 
is Apéry's constant, 
and 
are q-harmonic series, and the lower bounds are 2.
constant  | upper bound | reference |
 | 7.10320534 | Zeilberger and Zudilin (2020) |
 | 5.09541179 | Zudilin (2013) |
 | 3.57455391 | Marcovecchio (2009) |
 | 5.116201 | Bondareva et al. (2018) |
 | 5.513891 | Rhin and Viola (2001) |
 | 2.9384 | Matala-Aho et al. (2006) |
 | 2.4650 | Zudilin (2004) |
The bound for 
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 
be the complex conjugate roots of
 | (8) |
let 
be the positive real root, and let
 |  |  | (9) |
 |  |  | (10) |
 |  |  | (11) |
 |  |  | (12) |
then the bound is given by
 | (13) |
Alekseyev (2011) has shown that the question of the convergence of the Flint Hills series is related to the irrationality measure of 
, and in particular, convergence would imply 
, 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 
." 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 
and 
." Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-286, 1992.Hata, M. "Rational Approximations to 
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 
." Acta Arith. 92, 47-57, 2000.Marcovecchio, R. "The Rhin-Viola Method for 
." Acta Arith. 139, 147-184, 2009.Matala-Aho, T.; Väänänen, K.; and Zudilin, W. "New Irrationality Measures for 
-Logarithms." Math. Comput. 75, 879-889, 2006.Rhin, G. and Viola, C. "On a Permutation Group Related to 
." Acta Arith. 77, 23-56, 1996.Rhin, G. and Viola, C. "The Group Structure for 
." Acta Arith. 97, 269-293, 2001.Rukhadze, E.A. "A Lower Bound for the Rational Approximation of 
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 
."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 
." 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 
-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 
is at Most 7.103205334137...." 8 Jan 2020. https://arxiv.org/abs/1912.06345.Zudilin, V.V. "An Essay on the Irrationality Measures of 
and Other Logarithms." Chebyshevskiĭ Sb. 5, 49-65, 2004.Zudilin, V.V. "On the Irrationality Measure of 
." Russian Math. Surveys 68, 1133-1135, 2013.
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Cite this as:
Weisstein, Eric W. "Irrationality Measure."From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrrationalityMeasure.html
