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Über ein Problem der Gleichverteilungstheorie. (German) Zbl 0083.04401

The question of uniform distribution is considered for a sequence \(\{u_n\}\) of form
\[ \frac{1}{a_1}, \frac{2}{a_1}, \ldots, \frac{a_1-1}{a_1}, \frac{1}{a_2}, \ldots, \frac{a_2-1}{a_2},\ldots, \frac{1}{a_n}, \ldots, \frac{a_n-1}{a_n}, \ldots, \tag{1} \]
where \(\{a_n\}\) is a strictly increasing sequence of natural numbers. Let \(0 \le\alpha < \beta \le 1\) and let \(A(n)\) be the number of \(u_i\) with \(i\le n\) and \(\alpha\le u_i < \beta\). Let \(S\) be a continuous monotone function on \(x\ge 1\) such that whenever \(x\) is an integer \(S(x) = \displaystyle\sum_{\nu = 1}^x a_\nu - x\). Then we have the theorem
\[ A(n) = (\beta-\alpha) n + O(a_k), \]
where \(k = \langle S^{-1}(n)\rangle\). If \(a_k/k \to\infty\) and \([\alpha, \beta) \ne [0, 1)\) then
\[ A(n) = (\beta-\alpha) n + \Omega(a_k). \]
Here \(\langle \delta\rangle\) devotes the smallest integer \(l > \delta\), \(S^{-1}\) the inverse function to \(S\), and \(\Omega\) the negation of \(o\).
With the aid of two corollaries, we have the following criteria for uniform distribution.
I. If there is a function \(f\) such that \(S(f(n)) \ge n\) \((n\ge n_0)\), \(a_{\langle f(n)\rangle} = o(n)\), then the sequence (1) is uniformly distributed.
II. If \(a_k/k \to\infty\) and there is a function \(f\) with \(S(f(n)) \le n\) \((n\ge n_0)\), \(a_{[ f(n)]} \ne o(n)\), then (1) is not uniformly distributed.
The examples \(a_n = n\), \(a_n = a^n\) \((a > 1\), integral) and \(a_n = p_n\) \((n\)-th prime) are considered.

MSC:

11K06 General theory of distribution modulo \(1\)
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