**Author(s):**Rita Giuliano Antonini | Georges Grekos

**Journal:**Uniform Distribution Theory

ISSN 1336-913X

**Volume:**3;

**Issue:**2;

**Start page:**21;

**Date:**2008;

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**Keywords:**$\ | alpha-$density | $\ | alpha-$analytic density | logarithmic density | analytic density | tauberian theorem | slowly varying function | regular set

**ABSTRACT**

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 $\alpha > -1$.