Measure of irrationality

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 (cf. Linear independence, measure of; Transcendency, measure of). 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)