Every non-zero real number is the product of two normal numbers. This follows from the general fact that every number is the product of two numbers from a set ${\displaystyle X\subseteq \mathbb {R} ^{+}}$ if the complement of X has measure 0.

If x is normal in base b and a ≠ 0 is a rational number, then ${\displaystyle x\cdot a}$ is also normal in base b.

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If ${\displaystyle A\subseteq \mathbb {N} }$ is dense (for every ${\displaystyle \alpha <1}$ and for all sufficiently large n, ${\displaystyle |A\cap \{1,\ldots ,n\}|\geq n^{\alpha }}$) and ${\displaystyle a_{1},a_{2},a_{3},\ldots }$ are the base-b expansions of the elements of A, then the number ${\displaystyle 0.a_{1}a_{2}a_{3}\ldots }$, formed by concatenating the elements of A, is normal in base b (Copeland and Erdős 1946). From this it follows that Champernowne's number is normal in base 10 (since the set of all positive integers is obviously dense) and that the Copeland–Erdős constant is normal in base 10 (since the implies that the set of primes is dense).

prime number theorem

A sequence is normal every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005).

if and only if

A number is normal in base b if and only if it is simply normal in base b^k for all ${\displaystyle k\in \mathbb {Z} ^{+}}$. This follows from the previous block characterization of normality: Since the n^th block of length k in its base b expansion corresponds to the n^th digit in its base b^k expansion, a number is simply normal in base b^k if and only if blocks of length k appear in its base b expansion with equal frequency.

A number is normal if and only if it is simply normal in every base. This follows from the previous characterization of base b normality.

A number is b-normal if and only if there exists a set of positive integers ${\displaystyle m_{1}<m_{2}<m_{3}<\cdots }$ where the number is simply normal in bases b^m for all ${\displaystyle m\in \{m_{1},m_{2},\ldots \}.}$ No finite set suffices to show that the number is b-normal.

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All normal sequences are closed under finite variations: adding, removing, or changing a number of digits in any normal sequence leaves it normal. Similarly, if a finite number of digits are added to, removed from, or changed in any simply normal sequence, the new sequence is still simply normal.

finite

A finite-state gambler (a.k.a. finite-state ) is a finite-state machine over a finite alphabet ${\displaystyle \Sigma }$, each of whose states is labelled with percentages of money to bet on each digit in ${\displaystyle \Sigma }$. For instance, for an FSG over the binary alphabet ${\displaystyle \Sigma =\{0,1\}}$, the current state q bets some percentage ${\displaystyle q_{0}\in [0,1]}$ of the gambler's money on the bit 0, and the remaining ${\displaystyle q_{1}=1-q_{0}}$ fraction of the gambler's money on the bit 1. The money bet on the digit that comes next in the input (total money times percent bet) is multiplied by ${\displaystyle |\Sigma |}$, and the rest of the money is lost. After the bit is read, the FSG transitions to the next state according to the input it received. A FSG d succeeds on an infinite sequence S if, starting from $1, it makes unbounded money betting on the sequence; i.e., if

$${\displaystyle \limsup _{n\to \infty }d(S\upharpoonright n)=\infty ,}$$

where ${\displaystyle d(S\upharpoonright n)}$ is the amount of money the gambler d has after reading the first n digits of S (see limit superior).

martingale

A finite-state compressor is a finite-state machine with output strings labelling its , including possibly the empty string. (Since one digit is read from the input sequence for each state transition, it is necessary to be able to output the empty string in order to achieve any compression at all). An information lossless finite-state compressor is a finite-state compressor whose input can be uniquely recovered from its output and final state. In other words, for a finite-state compressor C with state set Q, C is information lossless if the function ${\displaystyle f:\Sigma ^{*}\to \Sigma ^{*}\times Q}$, mapping the input string of C to the output string and final state of C, is 1–1. Compression techniques such as Huffman coding or Shannon–Fano coding can be implemented with ILFSCs. An ILFSC C compresses an infinite sequence S if

$${\displaystyle \liminf _{n\to \infty }{\frac {|C(S\upharpoonright n)|}{n}}<1,}$$

where ${\displaystyle |C(S\upharpoonright n)|}$ is the number of digits output by C after reading the first n digits of S. The compression ratio (the limit inferior above) can always be made to equal 1 by the 1-state ILFSC that simply copies its input to the output.

state transitions

Champernowne constant

De Bruijn sequence

Infinite monkey theorem

The Library of Babel

"Normal number". MathWorld.

Weisstein, Eric W.