Lacunary distribution of sequences (Q803832)

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 \textit{H. Weyl} [Nachr. Ges. Wiss. Göttingen Math. Phys., 234-244 (1914)] and \textit{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)].