Difference between revisions of "Measure of irrationality"
From Encyclopedia of Mathematics
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| − | ''of a real number | + | {{TEX|done}}{{MSC|11J82}} |
| + | |||
| + | ''of a real number $\xi$'' | ||
The function | The function | ||
| − | + | $$L(\xi,H)=\min|h_1\xi+h_0|,$$ | |
| − | where the minimum is over all pairs | + | where the minimum is over all pairs $h_0,h_1$ of integral rational numbers such that |
| − | + | $$|h_0|,|h_1|\leq H,\quad |h_0|+|h_1|\neq0.$$ | |
| − | The concept of the measure of irrationality is a particular case of those of the | + | The concept of the measure of irrationality is a particular case of those of the [[Linear independence, measure of|measure of linear independence]] and the [[Transcendency, measure of|measure of transcendency]]. The measure of irrationality indicates how "well" the number $\xi$ can be approximated by rational numbers. For all real irrational numbers one has |
| − | < | + | $$L(\xi,H)<\frac1{\sqrt5}\frac1H,$$ |
| − | but for any | + | but for any $\epsilon>0$ and almost-all (in the sense of the Lebesgue measure) real numbers $\xi$, |
| − | + | $$L(\xi,H)>\frac{C}{H^{1+\epsilon}},$$ | |
| − | where | + | where $C=C(\epsilon,\xi)>0$. However, for any function $\phi$ with $\phi(H)\to0$ as $H\to\infty$ and $\phi(H)>0$, there exists a number $\xi_\phi$ such that for all $H\geq1$, |
| − | < | + | $$0<L(\xi_\phi,H)<\phi(H).$$ |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.Ya. Khinchin, "Continued fractions" , Univ. Chicago Press (1964) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.Ya. Khinchin, "Continued fractions" , Univ. Chicago Press (1964) (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 21:10, 14 December 2014
2020 Mathematics Subject Classification: Primary: 11J82 [MSN][ZBL]
of a real number $\xi$
The function
$$L(\xi,H)=\min|h_1\xi+h_0|,$$
where the minimum is over all pairs $h_0,h_1$ of integral rational numbers such that
$$|h_0|,|h_1|\leq H,\quad |h_0|+|h_1|\neq0.$$
The concept of the measure of irrationality is a particular case of those of the measure of linear independence and the measure of transcendency. The measure of irrationality indicates how "well" the number $\xi$ can be approximated by rational numbers. For all real irrational numbers one has
$$L(\xi,H)<\frac1{\sqrt5}\frac1H,$$
but for any $\epsilon>0$ and almost-all (in the sense of the Lebesgue measure) real numbers $\xi$,
$$L(\xi,H)>\frac{C}{H^{1+\epsilon}},$$
where $C=C(\epsilon,\xi)>0$. However, for any function $\phi$ with $\phi(H)\to0$ as $H\to\infty$ and $\phi(H)>0$, there exists a number $\xi_\phi$ such that for all $H\geq1$,
$$0<L(\xi_\phi,H)<\phi(H).$$
References
| [1] | A.Ya. Khinchin, "Continued fractions" , Univ. Chicago Press (1964) (Translated from Russian) |
How to Cite This Entry:
Measure of irrationality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_of_irrationality&oldid=11646
Measure of irrationality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_of_irrationality&oldid=11646
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article