%0 Journal Article
%T Comparison between lower and upper $ \alpha-$densities and lower and upper $ \alpha-$analytic densities
%A Rita Giuliano Antonini
%A Georges Grekos
%J Uniform Distribution Theory
%D 2008
%I Mathematical Institute of the Slovak Academy of Sciences
%X Let $\alpha$ be a real number, with $\alpha \geq - 1$. We prove a general inequality between the upper (resp. lower) $\alpha-$analytic density and the upper (resp. lower)$\alpha-$density of a subset $A$ of $\mathbb N^*$ (Proposition 2.1). Moreover, we prove by an example that the upper and the lower $\alpha$--densities and the lower and upper $\alpha$--analytic densities of $A$ do not coincide in general ({\it i.e.}, the inequalities proved in (2.1) may be strict). On the other hand, we identify a class of subsets of $\mathbb N^*$ for which these values do coincide in the case $\alpha > -1$.
%K $\
%K alpha-$density
%K $\
%K alpha-$analytic density
%K logarithmic density
%K analytic density
%K tauberian theorem
%K slowly varying function
%K regular set
%U http://www.boku.ac.at/MATH/udt/vol03/no2/GiuGr08-2.pdf