Measures of affinity of a sequence for a number

Abstract

We define strong and weak affinities of a number a for a sequence (xk) denoted by L (a,(xk)) and U (a, (xk)) respectively. We show U (a,(xk)) > 0 if and only if the number a is a statistical limit point of the sequence (xk). 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,(xk)) > 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?. © 2002 Taylor and Francis Group, LLC.