Difference between revisions of "Lagrange spectrum"
From Encyclopedia of Mathematics
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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|Markov spectrum problem]]). | 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|Markov spectrum problem]]). | ||
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| + | 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 | ||
Revision as of 11:29, 18 October 2014
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
How to Cite This Entry:
Lagrange spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_spectrum&oldid=11496
Lagrange spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_spectrum&oldid=11496