Vol. 18, No. 1, 1966

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ISSN: 0030-8730
On real numbers having normality of order k

Calvin T. Long

Vol. 18 (1966), No. 1, 155–160
Abstract

This paper contains three theorems concerning real numbers having normality of order k. The first theorem gives a simple construction of a periodic decimal having normality of order k to base r. After introducing the notion of c-uniform distribution modulo one, we prove in the second theorem that α has normality of order k to base r if and only if the function αrx is rk-uniformly distributed modulo one. In the third theorem we show that α has normality of order k to base r if and only if, for every integer b and every positive integer t k,

   N-(b,n)   −t
lim    r   = r

where N(b,n) is the number of integers x with 1 x n for which

  x           t
[αr ] ≡ b (mod r).

Mathematical Subject Classification
Primary: 10.33
Milestones
Received: 4 February 1965
Published: 1 July 1966
Authors
Calvin T. Long