On Strong Uniform Distribution II

Abstract.

 Let ? be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers an sequence if for every f in ? we have

almost everywhere with respect to Lebesgue measure. Here, for a real number y we have used to denote the fractional part of y. For a finite set A we use to denote its cardinality. In this paper we show that for strictly increasing sequences of natural numbers and , both of which are sequences for all , if there exists such that

then the sequence of products of pairs of elements in a and b once ordered by size is also an sequence.