Difference between revisions of "Lagrange spectrum"
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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 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 | Given positive real $\alpha$, define the homogeneous approximation constant, or Lagrange constant, $\lambda(\alpha)$, to be the supremum of values $c$ for which | ||
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$$ | $$ | ||
has infinitely many solutions in coprime integers $p,q$. The Lagrange spectrum $L$ is the set of all values taken by the function $\lambda$. | 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 | ||
| + | $$ | ||
| + | x > 4 + \frac{253589820 + 283748\sqrt{462}}{491993569} \sim 4.5278395661 \ldots | ||
| + | $$ | ||
====References==== | ====References==== | ||
Revision as of 13:27, 18 October 2014
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).
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 $$ x > 4 + \frac{253589820 + 283748\sqrt{462}}{491993569} \sim 4.5278395661 \ldots $$
References
| [a1] | 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) |
| [a2] | Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2 |
Lagrange spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_spectrum&oldid=33789