Normal number

2020 Mathematics Subject Classification: Primary: 11K16 [MSN][ZBL]

A real number $\alpha$, $0 < \alpha < 1$, having the following property: For every natural number $s$, any given $s$-tuple $\delta = (\delta_1,\ldots,\delta_s)$ consisting of the symbols $0,1,\ldots,g-1$ appears with asymptotic frequency $1/g^s$ in the sequence $$\label{eq:a1} \alpha_1,\ldots,\alpha_n,\ldots $$ obtained from the expansion of $\alpha$ as an infinite fraction in base $g$, $$ \alpha = \frac{\alpha_1}{g} + \cdots + \frac{\alpha_n}{g^n} + \cdots \ . $$

In more detail, let $g>1$ be a natural number and let $$\label{eq:a2} (\alpha_1,\ldots,\alpha_s), (\alpha_2,\ldots,\alpha_{s+1}), \ldots $$ be the infinite sequence of $s$-tuples corresponding to \ref{eq:a1}. Let $N(n,\delta)$ denote the number of occurrences of the tuple $\delta = (\delta_1,\ldots,\delta_s)$ among the first $n$ tuples of \ref{eq:a2}. The number $$ \alpha = \frac{\alpha_1}{g} + \cdots + \frac{\alpha_n}{g^n} + \cdots \ . $$ is said to be normal if for any number $s$ and any given $s$-tuple $\delta$ consisting of the symbols $0,\ldots,g-1$, $$ \lim_{n \rightarrow \infty} \frac{N(n,s)}{ n } = \frac{1}{g^s} \ . $$

The concept of a normal number was introduced for $g=10$ by E. Borel (see [B], [B2], p. 197). He called a real number $\alpha$ weakly normal to the base $g$ if $$ \lim_{n \rightarrow \infty} \frac{N(n,\delta)}{n} = \frac{1}{g} $$ where $N(n,\delta)$ is the number of occurrences of $\delta$, $0 \le \delta \le g-1$, among the first $n$ terms of the sequences $\alpha_1,\alpha_2,\ldots$ and normal if $\alpha, g\alpha, g^2\alpha, \ldots$ are weakly normal to the bases $g, g^2, \ldots$. He also showed that for a normal number $$ \lim_{n \rightarrow \infty} \frac{N(n,s)}{ n } = \frac{1}{g^s} $$ for any $s$ and any given $s$-tuple $\delta = (\delta_1,\ldots,\delta_s)$. Later it was proved (see [Pi], [NZ], and also [Po]) that the last relation is equivalent to Borel's definition of a normal number.

A number $\alpha$ is called absolutely normal if it is normal with respect to every base $g$. The existence of normal and absolutely-normal numbers was established by Borel on the basis of measure theory. The construction of normal numbers in an explicit form was first achieved in [C]. Earlier (see [S], [L]) an effective procedure for constructing normal numbers was indicated. For other methods for constructing normal numbers and for connections between the concepts of normality and randomness see [Po].

Uniform distribution of the fractional parts $\{ \alpha g^x \}$, $x = 0, 1, \ldots$ on the interval $[0,1]$ is equivalent to $\alpha$ being normal.

References

[B]	E. Borel, "Les probabilités dénombrables et leurs applications arithmétiques" Rend. Circ. Math. Palermo , 27 (1909) pp. 247–271 Zbl 40.0283.01
[B2]	E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928) MR0033328 Zbl 54.0327.02
[Pi]	S. Pillai, "On normal numbers" Proc. Indian Acad. Sci. Sect. A , 12 (1940) pp. 179–184 MR0002324 Zbl 0025.30802 Zbl 66.1212.02
[NZ]	I. Niven, H. Zuckerman, "On the definition of normal numbers" Pacific J. Math. , 1 (1951) pp. 103–109 MR0044560 Zbl 0042.26902
[C]	D.G. Champernowne, "The construction of decimals normal in the scale of ten" J. London Math. Soc. , 8 (1933) pp. 254–260 Zbl 0007.33701 Zbl 59.0214.01
[S]	W. Sierpiński, "Démonstration élémentaire d'un théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre" Bull. Soc. Math. France , 45 (1917) pp. 127–132 MR0073664 MR0055398 MR0021058 MR1550055
[L]	H. Lebesgue, "Sur certaines démonstrations d'existence" Bull. Soc. Math. France , 45 (1917) pp. 132–144 MR1504765
[Po]	A.G. Postnikov, "Arithmetic modelling of random processes" Trudy Mat. Inst. Steklov. , 57 (1960) (In Russian)

Comments

Almost all numbers (with respect to Lebesgue measure) are normal with respect to every base $g$ (see e.g. Theorem 8.11 in [N] or section 9.13 of [HW]). It is not known whether familiar numbers like $\sqrt2,\,e,\,\pi$ are normal or not. Normal numbers are potentially interesting in the context of random number generators. A normal number to a base $g$ is necessarily irrational. The weakly-normal number (to base $10$) $0\cdot12345678901234567890\ldots$ is of course rational. The Champernowne number $x = 0\cdot1234567891011121314\ldots$, obtained as $x = 0 \cdot \alpha_1 \alpha_2 \alpha_3 \ldots$ where $\alpha_i$ stands for the group of digits representing $i$ to base $10$, is normal to base $10$ [C]. The same recipe works to obtain normal numbers to any given base.

References

[HW]	Hardy, G. H.; Wright, E. M. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press (2008) [1938] ISBN 978-0-19-921986-5 Zbl 1159.11001
[N]	I. Niven, "Irrational numbers" , Math. Assoc. Amer. (1956) MR1570844 MR0080123 Zbl 0070.27101

Comments

The example of Champernowne's number as an explicitly normal number in base 10 was generalised by Copeland and Erdős [CE] who showed that if $a_n$ is an increasing sequence of natural numbers with the property that for every $\theta > 1$ then $a_n < n^\theta$ for sufficiently large $n$, then the number $0 \cdot \alpha_1 \alpha_2 \ldots$ is normal in base $g$ where $\alpha_n$ is the base $g$ expression of $a_n$. See also [B] pp.86-87.

References

[CE]	Copeland, A. H.; Erdős, P. "Note on normal numbers", Bulletin of the American Mathematical Society 52 (1946) 857–860, DOI 10.1090/S0002-9904-1946-08657-7 Zbl 0063.00962
[B]	Bugeaud, Yann. Distribution modulo one and Diophantine approximation, Cambridge Tracts in Mathematics 193, Cambridge: Cambridge University Press (2012) ISBN 978-0-521-11169-0, Zbl 1260.11001