Journal of Inequalities and Applications
Volume 2005 (2005), Issue 3, Pages 319-327
On strong uniform distribution IV
Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, UK
Received 24 January 2003
Copyright © 2005 R. Nair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.