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,
where N(b,n) is the number of integers x with 1 ≦ x ≦ n for which
![x t
[αr ] ≡ b (mod r).](a151x.png)
