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Lacunary distribution of sequences. (English) Zbl 0726.40002
For the known convergence spaces of sequences of real numbers with the evident properties $\vert C\sb 1\vert \subset C\sb 1$, $\vert C\sb{\theta}\vert \subset C\sb{\theta}$, $\vert AC\vert \subset AC$ and $\vert AC\sb{\theta}\vert \subset AC\sb{\theta}$, the authors first prove the theorems (1) $\vert AC\sb{\theta}\vert \Leftrightarrow \vert AC\vert$ for every $\theta$. (2a) For some $\theta$, $AC\sb{\theta}\neg \Rightarrow I\infty$. (b) For every $\theta$, $AC\sb{\theta}\cap I\infty \Leftrightarrow AC.$ They then define the concept of uniformity and well distributedness modulo 1, of the sequence of real numbers over the lacunary sequence $\theta$ on the lines of {\it H. Weyl} [Nachr. Ges. Wiss. Göttingen Math. Phys., 234-244 (1914)] and {\it G. M. Petersen} [Quart. J. Math., Oxford II. Ser. 7, 188-191 (1956; Zbl 0072.273)] and prove two theorems similar to their own on uniformity asymptotic distribution functions [Ph. D. Thesis submitted to Sambalpur University (1982)].

40A05Convergence and divergence of series and sequences
40E15Lacunary inversion theorems