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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 $|A{C}_{\theta }|\subset A{C}_{\theta }$, the authors first prove the theorems (1) $|A{C}_{\theta }|⇔|AC|$ for every $\theta$. (2a) For some $\theta$, $A{C}_{\theta }¬⇒I\infty$. (b) For every $\theta$, $A{C}_{\theta }\cap I\infty ⇔AC·$

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