On Weighted Uniform Density

Uniform density (also known as Banach density) was often used in various branches of mathematics, in particular in number theory and ergodic theory. Several characterizations of uniform density were given in the paper [G´alikov´a, Z. – L´aszl´o, B. – Sˇal´at, T.:Remarks on uniform density of sets of integers, Acta Acad. Paed. Agriensis, 2002]. The notion of uniform density was recently generalized to weighted uniform density in the paper [Giuliano Antonini, R. – Grekos, G.: Weighted uniform den- sities, Journal de th´eorie des nombres de Bordeaux, 2007]. Some sufficient con- ditions for the existence of upper and lower weighted uniform density for every subset of the set of integers have been obtained in this paper. We show that for positive weights the upper and lower weighted uniform den- sity always exist. We also show that the alternative characterizations of the uni- form density remain valid for the weighted uniform density as well. Moreover, we investigate related characterizations of the upper and lower weighted uniform density.