Difference between revisions of "Linear independence, measure of"
From Encyclopedia of Mathematics
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| − | + | ''of numbers $\alpha_1,\ldots,\alpha_n$'' | |
| − | + | {{MSC|11J}} | |
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| + | 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 | It is known that | ||
| − | + | $$ | |
| − | < | + | L(\alpha_1,\ldots,\alpha_n|H) < (|a_1| + \cdots + |a_n|) H^{\tau(n-1)} |
| − | + | $$ | |
| − | where | + | 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=14591
Linear independence, measure of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_independence,_measure_of&oldid=14591
This article was adapted from an original article by Yu.V. Nesterenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article