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Difference between revisions of "Linear independence, measure of"

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''of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593001.png" />''
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{{TEX|done}}
  
The function
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''of numbers $\alpha_1,\ldots,\alpha_n$''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593002.png" /></td> </tr></table>
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{{MSC|11J}}
 
 
where the minimum is taken over all possible sets of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593003.png" /> satisfying the conditions
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593004.png" /></td> </tr></table>
 
  
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The function
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$$
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L(\alpha_1,\ldots,\alpha_n|H) = L(H) = \min |a_1 \alpha_1 + \cdots + a_n \alpha_n|
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$$
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where the minimum is taken over all possible sets of integers $a_1,\ldots,a_n$, not all zero, satisfying the conditions
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$$
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|a_i| \le H
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$$
 
It is known that
 
It is known that
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593005.png" /></td> </tr></table>
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L(\alpha_1,\ldots,\alpha_n|H) < (|a_1| + \cdots + |a_n|) H^{\tau(n-1)}
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593006.png" /> if all the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593007.png" /> are real, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593008.png" /> otherwise. To obtain lower bounds of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l0593009.png" /> with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l05930010.png" /> for specific sets of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059300/l05930011.png" /> is one of the problems of the theory of [[Diophantine approximations|Diophantine approximations]].
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where $\tau=1$ if all the numbers $\alpha_1,\ldots,\alpha_n$ are real, and $\tau=\frac12$ otherwise. To obtain lower bounds on $L(\alpha_1,\ldots,\alpha_n|H)$ with respect to the parameter $H$ for specific sets of numbers $\alpha_1,\ldots,\alpha_n$ is one of the problems of the theory of [[Diophantine approximations]].

Latest revision as of 19:48, 20 November 2014


of numbers $\alpha_1,\ldots,\alpha_n$

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

The function $$ L(\alpha_1,\ldots,\alpha_n|H) = L(H) = \min |a_1 \alpha_1 + \cdots + a_n \alpha_n| $$ where the minimum is taken over all possible sets of integers $a_1,\ldots,a_n$, not all zero, satisfying the conditions $$ |a_i| \le H $$ It is known that $$ L(\alpha_1,\ldots,\alpha_n|H) < (|a_1| + \cdots + |a_n|) H^{\tau(n-1)} $$ where $\tau=1$ if all the numbers $\alpha_1,\ldots,\alpha_n$ are real, and $\tau=\frac12$ otherwise. To obtain lower bounds on $L(\alpha_1,\ldots,\alpha_n|H)$ with respect to the parameter $H$ for specific sets of numbers $\alpha_1,\ldots,\alpha_n$ is one of the problems of the theory of Diophantine approximations.

How to Cite This Entry:
Linear independence, measure of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_independence,_measure_of&oldid=34658
This article was adapted from an original article by Yu.V. Nesterenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article