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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 $$| C_ 1| \subset C_ 1$$, $$| C_{\theta}| \subset C_{\theta}$$, $$| AC| \subset AC$$ and $$| AC_{\theta}| \subset AC_{\theta}$$, the authors first prove the theorems (1) $$| AC_{\theta}| \Leftrightarrow | AC|$$ for every $$\theta$$. (2a) For some $$\theta$$, $$AC_{\theta}\neg \Rightarrow I\infty$$. (b) For every $$\theta$$, $$AC_{\theta}\cap I\infty \Leftrightarrow AC.$$
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 H. Weyl [Nachr. Ges. Wiss. Göttingen Math. Phys., 234-244 (1914)] and 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)].

##### MSC:
 40A05 Convergence and divergence of series and sequences 40E15 Lacunary inversion theorems
##### Keywords:
lacunary distribution; convergence spaces
Zbl 0072.273