Measures of affinity of a sequence for a number

Abstract

We define strong and weak affinities of a number a for a sequence
(x_k)
denoted by L(a,(x_k))
and U(a,(x_k)) respectively. We show U(a,(x_k)) 〉 0
if
and only if the number a is a statistical limit point of the sequence
(x_k).
We consider the distribution
of sequences with positive weak and strong measures of affinity within the
space l^∞ of bounded sequences. The main result is that
the set of bounded sequences with U(a,(x_k)) 〉 0,
that is, the set of sequences with statistical limit points, is a dense subset
in l^∞ of the first category. We also show
the set of sequences with positive strong affinities is a nowhere dense subset
of l^∞.



Mathematics Subject
Classification (2000): 40A05



Quaestiones Mathematicae 25 (2002), 473-481