For the known convergence spaces of sequences of real numbers with the evident properties , , and , the authors first prove the theorems (1) for every . (2a) For some , . (b) For every ,
They then define the concept of uniformity and well distributedness modulo 1, of the sequence of real numbers over the lacunary sequence 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)].