Absolutely Normal

A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in  are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).

The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).

REFERENCES:

Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.

Borel, E. "Les probabilités dénombrables et leurs applications arithmétiques." Rend. Circ. Mat. Palermo 27, 247-271, 1909.

Borel, E. Leçons sur la théorie de fonctions. Paris, pp. 197-198, 1922.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 143, 2003.

Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974.

Niven, I. M. Irrational Numbers. New York: Wiley, 1956.

Schmidt, W. "Über die Normalität von Zahlen zu verschiedenen Basen." Acta Arith. 7, 299-309, 1962.

Sierpiński, W. "Démonstration élémentaire d'un théorème de M. Borel sue les nombres absolutment normaux et détermination effective d'un tel nombre." Bull. Soc. Math. France 45, 125-144, 1917.

Sierpiński, W. and Schinzel, A. Elementary Theory of Numbers, 2nd Eng. ed. Amsterdam, Netherlands: North-Holland, 1988.

Stoneham, R. "A General Arithmetic Construction of Transcendental Non-Liouville Normal Numbers from Rational Functions." Acta Arith. 16, 239-253, 1970.