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.