On strong uniform distribution IV

Abstract

Let be a strictly increasing sequence of natural numbers and let be a space of Lebesgue measurable functions defined on. Let denote the fractional part of the real number. We say that is an sequence if for each we set, then, almost everywhere with respect to Lebesgue measure. Let. In this paper, we show that if is an for, then there exists such that if denotes,. We also show that for any sequence and any nonconstant integrable function on the interval,, almost everywhere with respect to Lebesgue measure.