On strong uniform distribution IV

• Main Author: Nair R
• Format: Article
• Language: English
• Published: SpringerOpen 2005-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2005/639193

<p/> <p>Let <inline-formula><graphic file="1029-242X-2005-639193-i1.gif"/></inline-formula> be a strictly increasing sequence of natural numbers and let <inline-formula><graphic file="1029-242X-2005-639193-i2.gif"/></inline-formula> be a space of Lebesgue measurable functions defined on <inline-formula><graphic file="1029-242X-2005-639193-i3.gif"/></inline-formula>. Let <inline-formula><graphic file="1029-242X-2005-639193-i4.gif"/></inline-formula> denote the fractional part of the real number <inline-formula><graphic file="1029-242X-2005-639193-i5.gif"/></inline-formula>. We say that <inline-formula><graphic file="1029-242X-2005-639193-i6.gif"/></inline-formula> is an <inline-formula><graphic file="1029-242X-2005-639193-i7.gif"/></inline-formula> sequence if for each <inline-formula><graphic file="1029-242X-2005-639193-i8.gif"/></inline-formula> we set <inline-formula><graphic file="1029-242X-2005-639193-i9.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2005-639193-i10.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2005-639193-i11.gif"/></inline-formula>, almost everywhere with respect to Lebesgue measure. Let <inline-formula><graphic file="1029-242X-2005-639193-i12.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2005-639193-i13.gif"/></inline-formula>. In this paper, we show that if <inline-formula><graphic file="1029-242X-2005-639193-i14.gif"/></inline-formula> is an <inline-formula><graphic file="1029-242X-2005-639193-i15.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-2005-639193-i16.gif"/></inline-formula>, then there exists <inline-formula><graphic file="1029-242X-2005-639193-i17.gif"/></inline-formula> such that if <inline-formula><graphic file="1029-242X-2005-639193-i18.gif"/></inline-formula> denotes <inline-formula><graphic file="1029-242X-2005-639193-i19.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-639193-i20.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2005-639193-i21.gif"/></inline-formula>. We also show that for any <inline-formula><graphic file="1029-242X-2005-639193-i22.gif"/></inline-formula> sequence <inline-formula><graphic file="1029-242X-2005-639193-i23.gif"/></inline-formula> and any nonconstant integrable function <inline-formula><graphic file="1029-242X-2005-639193-i24.gif"/></inline-formula> on the interval <inline-formula><graphic file="1029-242X-2005-639193-i25.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-639193-i26.gif"/></inline-formula>, almost everywhere with respect to Lebesgue measure.</p>