Namespaces
Variants
Actions

Difference between revisions of "Lagrange spectrum"

From Encyclopedia of Mathematics
Jump to: navigation, search
(numerical values, cite Finch (2003))
m (→‎References: isbn link)
 
(3 intermediate revisions by one other user not shown)
Line 1: Line 1:
 +
{{TEX|done}}{{MSC|11J06}}
 +
 
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]]).
 
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]]).
  
Line 11: Line 13:
 
x > 4 + \frac{253589820 + 283748\sqrt{462}}{491993569} \sim 4.5278395661 \ldots
 
x > 4 + \frac{253589820 + 283748\sqrt{462}}{491993569} \sim 4.5278395661 \ldots
 
$$
 
$$
 +
this is ''Freiman's constant''.
  
 
====References====
 
====References====
 
<table>
 
<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)</TD></TR>
+
<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</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>
 
</table>
 
[[Category:Number theory]]
 

Latest revision as of 16:51, 23 November 2023

2020 Mathematics Subject Classification: Primary: 11J06 [MSN][ZBL]

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 $$ x > 4 + \frac{253589820 + 283748\sqrt{462}}{491993569} \sim 4.5278395661 \ldots $$ this is Freiman's constant.

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) Zbl 0362.10027
[a2] Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
How to Cite This Entry:
Lagrange spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_spectrum&oldid=33789