On Small Sets of Distribution Functions of Ratio Block Sequences

There are various methods how to describe and characterize distribution of elements of sets of positive integers. One of the most interesting is that using the set of all distribution functions of the corresponding ratio block sequence introduced in [Strauch, O.—Tóth, J.T.: Publ. Math. Debrecen 58 (2001), no. 4, 751–778]. In the present paper we give some sufficient conditions under which this set is small in a metric sense. As a corollary we obtain a new characterization of the case of asymptotic distribution.