Note on normal numbers
1957; Mathematical Sciences Publishers; Volume: 7; Issue: 2 Linguagem: Inglês
10.2140/pjm.1957.7.1163
ISSN1945-5844
Autores ResumoIntroduction* Let a be a real number with fractional part .awhen written to base r.Let Y n denote the block of the first n digits in this representation and let N(d, Y n ) denote the number of occurrences of the digit ίί in Γ B .The number a is said to be simply normal to base r if lim for each of the r distinct choices of d. α is said to be normal to base r if each of the numbers a, roc, r 2 a, are simply normal to each of the bases r, r 2 , r 3 , .These definitions, due to Emile Borel [1], were introduced in 1909.In 1940 S. S. Pillai [3] showed that a necessary and sufficient condition that a be normal to base r is that it be simply normal to each of the bases r, r 2 , r 3 , , thus considerably reducing the number of conditions needed to imply normality.The purpose of the present note is to show that a is normal to base r if and only if there exists a set of positive integers m 1 < m % < ra 3 < such that a is simply normal to base r m * for each il>l, and also to show that no finite set of m's will suffice.Notation-We make use of the following additional conventions.If B k is any block of k digits to base r, N(B k , Y n ) will denote the number of occurrences of B k in Y n and N^B^, Y n ) will denote the number of occurrences of B k starting in positions congruent to i modulo k in Y n .The term " relative frequency " will denote the asymptotic frequency with which an event occurs.For example, B k occurs in (α), the fractional part of a, with relative frequency r~k if limiV^, Y n )ln=r~k.Proof of the theorems.The following lemmas are easily proved.m LEMMA 1.If lim Σ/*(*&)=1 an ^ if lim inf/^ra)^ 1/m for i=l, 2, , m; then lim/ έ (%)=l/m for each i.LEMMA 2. The real number a is simply normal to base r h if and
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