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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>
== Khinchin's theorem on metric Diophantine approximation and extensions == <!-- [[Khinchin's theorem on Diophantine approximations]] links here -->
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