On the definition of normal numbers
1951; Mathematical Sciences Publishers; Volume: 1; Issue: 1 Linguagem: Inglês
10.2140/pjm.1951.1.103
ISSN1945-5844
Autores Tópico(s)Mathematics and Applications
ResumoIntroduction, Let R be a real number with fractional part .^^2^3* * ' w hen written to scale r.Let N(b,n) denote the number of occurrences of the digit b in the first n places.The number R is said to be simply normal to scale r if (1) for each of the r possible values of b R is said to be normal to scale r if all the numbers R,rR,r 2 R, are simply normal to all the scales r,r 2 ,r 3 , .These definitions, for r = 10, were introduced by Emile Borel [l], who stated (p.261) that "la propriete' caracte'ristique" of a normal number is the following: that for any sequence B whatsoever of v specified digits, we have where N(B,n) stands for the number of occurrences of the sequence B in the first n decimal places.Several writers, for example Champernowne [2], Koksma [3, p. 116], and Cope land and Erdos [4], have taken this property (2) as the definition of a normal number.Hardy and Wright [5, p. 124] state that property ( 2) is equivalent to the definition, but give no proof.It is easy to show that a normal number has property (2), but the implication in the other direction does not appear to be so obvious.If the number R has property (2) then any sequence of digitsappears with the appropriate frequency, but will the frequencies all be the same for i -1,2, , v if we count only those occurrences of B such that bγ is an i, ί + v, i + 2v 9 -th digit?It is the purpose of this note to show that this is
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