Liouville number

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

A real number $\alpha$ such that for any $\nu\geq1$ the inequality

$$\left|\alpha-\frac pq\right|<q^{-\nu}$$

has infinitely many integer solutions $p$ and $q$ satisfying the conditions $q>0$, $(p,q)=1$. The fact that a Liouville number is transcendental (cf. Transcendental number) follows from the Liouville theorem (cf. Liouville theorems). These numbers were studied by J. Liouville [1].

Examples of Liouville numbers are:

$$\alpha_1=\sum_{n=1}^\infty2^{-n!},$$

$$\alpha_2=\sum_{n=1}^\infty(-1)^n2^{-3^n},$$

$$\alpha_3=\sum_{n=1}^\infty(10^{n!})^{-1}.$$

References

• [1] J. Liouville, "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques", C.R. Acad. Sci. Paris, 18 (1844) pp. 883–885
• [2] A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
• [a1] O. Perron, "Die Lehre von den Kettenbrüchen", 1, Teubner (1977) Sect. 35
• [a2] O. Perron, "Irrationalzahlen", Chelsea, reprint (1948)