On strong uniform distribution II

Authors
Citation
R. Nair, On strong uniform distribution II, MONATS MATH, 132(4), 2001, pp. 341-348
Citations number
14
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
MONATSHEFTE FUR MATHEMATIK
ISSN journal
0026-9255 → ACNP
Volume
132
Issue
4
Year of publication
2001
Pages
341 - 348
Database
ISI
SICI code
0026-9255(2001)132:4<341:OSUDI>2.0.ZU;2-L
Abstract
Let A be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers a = (a(r))(r=1)(infinity ) an A* sequence if for every f in A we have lim(n --> infinity) (1)/(n) Sigma (n)(r=1) f(<a(r)x >) = integral (1)(0) f( t)dt, almost everywhere with respect to Lebesgue measure. Here, for a real number ), we have used <y > to denote the fractional part of y. For a finite set A we use /A/ to denote its cardinality. In this paper we show that for stric tly increasing sequences of natural numbers a = (a(r))(r=1)(infinity) and b = (b(s))(s=1)(infinity), both of which are (L-P)* sequences for all p >1, if there exists C > 0 such that /{r : a(r) less than or equal to u} // {s : b(x) less than or equal to u}/ less than or equal to C/{(r, s) : a(r)b(x) less than or equal tou}/, (u = 1 , 2, ...) then the sequence of products of pairs of elements in a and b once ordered by size is also an (L-p)* sequence.