Journal of Inequalities and Applications
Volume 2005 (2005), Issue 3, Pages 319-327

On strong uniform distribution IV

R. Nair

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 a=(ai)i=1 be a strictly increasing sequence of natural numbers and let 𝒜 be a space of Lebesgue measurable functions defined on [0,1). Let {y} denote the fractional part of the real number y. We say that a is an 𝒜 sequence if for each f𝒜 we set AN(f,x)=(1/N)i=1Nf({aix})(N=1,2,), then limN  AN(f,x)=01f(t)dt, almost everywhere with respect to Lebesgue measure. Let Vq(f,x)=(N=1|AN+1(f,x)AN(f,x)|q)1/q(q1). In this paper, we show that if a is an (Lp) for p>1, then there exists Dq>0 such that if fp denotes (01|f(x)|pdx)1/p, Vq(f,·)qDqfp(q>1). We also show that for any (L1) sequence a and any nonconstant integrable function f on the interval [0,1), V1(f,x)=, almost everywhere with respect to Lebesgue measure.