View source for Lagrange spectrum

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The set of Lagrange constants in the problem of rational approximation to real numbers. The Lagrange spectrum is strictly contained in the Markov spectrum (see [[Markov spectrum problem|Markov spectrum problem]]).

Given positive real $\alpha$, define the homogeneous approximation constant, or Lagrange constant, $\lambda(\alpha)$, to be the supremum of values $c$ for which
$$
\left\vert{\alpha -\frac{p}{q} }\right\vert < \frac{1}{c q^2}
$$
has infinitely many solutions in coprime integers $p,q$.  The Lagrange spectrum $L$ is the set of all values taken by the function $\lambda$.

The smallest number in $L$ is $\sqrt{5}$.  The Lagrange and the Markov spectrum agree in the range [2,3].  Each spectrum contains the infinite half-line 
$$
x > 4 + \frac{253589820 + 283748\sqrt{462}}{491993569} \sim 4.5278395661 \ldots
$$
this is ''Freiman's constant''.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A.V. Malyshev, "Markov and Lagrange spectra (a survey of the literature)" ''Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.'' , '''67''' (1977) pp. 5–38 (In Russian) {{ZBL|0362.10027}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> Steven R. Finch, ''Mathematical Constants'', Cambridge University Press (2003) {{ISBN|0-521-81805-2}}  {{ZBL|1054.00001}}</TD></TR>
</table>