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On the uniform distribution of numbers mod. 1. (English) Zbl 0048.03302
Let $$\{\lambda_n\}$$ be a real sequence satisfying $$\lambda_1\ge \lambda_2\ge \lambda_3\ge \cdots >0$$, $$\sum_{n=1}^\infty \lambda_n=\infty$$. Let $$I$$ be any subinterval of $$[0, 1]$$, say of length $$\vert I\vert$$, and let $$\varphi(x)$$ be 1 or 0 according as to whether $$x\in I$$ or $$x\notin I$$. The real sequence $$\{x_n\}$$ is said to be $$\{\lambda_n\}$$-uniformly distributed (mod 1) if for all $$I$$, $\lim_{n\to\infty} \frac{\lambda_1\varphi(\bar x_1)+\ldots+\lambda_n\varphi(\bar x_n)}{\lambda_1+\ldots+\lambda_n} = \vert I\vert,$ where $$\bar x = x - [x]$$. The author proves several theorems analogous to those of Weyl, van der Corput, and Fejér for the case of ordinary uniform distribution (mod 1). By way of example, $$a n^\sigma (\log n)^{\sigma_1}$$ is $$\{1/n\}$$-uniformly distributed if $$a>0$$ and either $$\sigma > 0$$ or $$\sigma = 0$$ and $$\sigma_1 > 0$$.
Show Scanned Page ##### MSC:
 11K06 General theory of distribution modulo $$1$$ 11J71 Distribution modulo one
##### Keywords:
uniform distribution
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