Lagrange Number (Rational Approximation)

Hurwitz's Irrational Number Theorem gives the best rational approximation possible for an arbitrary irrational number $\alpha$ as

$$\left\vert{\alpha-{p\over q}}\right\vert<{1\over L_n q^2}.$$

The $L_n$ are called Lagrange numbers and get steadily larger for each ``bad'' set of irrational numbers which is excluded.

$n$	Exclude	$L_n$
1	none	$\sqrt{5}$
2	$\phi$	$\sqrt{8}$
3	$\sqrt{2}$	${\sqrt{221}\over 5}$

Lagrange numbers are of the form

$$\sqrt{9-{4\over m^2}},$$

where $m$ is a Markov Number. The Lagrange numbers form a Spectrum called the Lagrange Spectrum.

See also Hurwitz's Irrational Number Theorem, Liouville's Rational Approximation Theorem, Liouville-Roth Constant, Markov Number, Roth's Theorem, Spectrum Sequence, Thue-Siegel-Roth Theorem

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187-189, 1996.