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 \(\sigma_1 > 0\).


11K06 General theory of distribution modulo \(1\)
11J71 Distribution modulo one
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