Uniformly maldistributed sequences in a strict sense

Abstract

It is shown that the following three limits

$$\begin{gathered} \mathop {\lim }\limits_{n \to \infty } \frac{1}{{N^2 }}\sum\limits_{m,n = 1}^N {|x_m - x_n | = 0,} \hfill \\ \mathop {\lim \inf }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {x_n = 0,\mathop {\lim \sup }\limits_{N \to \infty } } \frac{1}{N}\sum\limits_{n = 1}^N {x_n = 1} \hfill \\ \end{gathered} $$

are a necessary and sufficient condition for the given sequence ω=(x _n) ^=1/∞ _n ⊄[0, 1] to have its only distribution functions be all one-jump functions. As an application, such sequences can also be used in deriving estimates of maxf for continuous functionsf defined in [0, 1].