Lagrange spectrum

The set of Lagrange constants in the problem of rational approximation to real numbers. The Lagrange spectrum is contained in the Markov spectrum (see 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 $\lambda$.

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2