Editing Diophantine approximation (section)

===Lagrange spectrum ===
{{main|Markov spectrum}}
As said above, the constant in Borel's theorem may not be improved, as shown by [[Adolf Hurwitz]] in 1891.<ref>{{harvnb|Hardy|Wright|1979|p=164}}</ref>
Let <math>\phi = \tfrac{1+\sqrt{5}}{2}</math> be the [[golden ratio]].
Then for any real constant ''c'' with <math>c > \sqrt{5}\;</math> there are only a finite number of rational numbers {{math|''p''/''q''}} such that 
:<math>\left|\phi-\frac{p}{q}\right| < \frac{1}{c\, q^2}.</math>

Hence an improvement can only be achieved, if the numbers which are equivalent to <math>\phi</math> are excluded. More precisely:<ref>{{harvnb|Cassels|1957|p=11}}</ref><ref>{{harvnb|Hurwitz|1891}}</ref>
For every irrational number <math>\alpha</math>, which is not equivalent to <math>\phi</math>, there are infinite many fractions <math>\tfrac{p}{q}\;</math> such that

: <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{8} q^2}.</math>

By successive exclusions — next one must exclude the numbers equivalent to <math>\sqrt 2</math> — of more and more classes of equivalence, the lower bound can be further enlarged.
The values which may be generated in this way are ''Lagrange numbers'', which are part of the [[Markov spectrum|Lagrange spectrum]]. 
They converge to the number 3 and are related to the [[Markov number]]s.<ref>{{harvnb|Cassels|1957|p=18}}</ref><ref>See [http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf Michel Waldschmidt: ''Introduction to Diophantine methods irrationality and transcendence''] {{Webarchive|url=https://web.archive.org/web/20120209111526/http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf |date=2012-02-09 }}, pp 24–26.</ref>