Abstract densities and ideals of sets
Volume 185 / 2018
Acta Arithmetica 185 (2018), 301-313
MSC: Primary 11B05; Secondary 03E05.
DOI: 10.4064/aa170417-13-12
Published online: 27 July 2018
Abstract
Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density.
We answer a question posed by G. Grekos in 2013, and prove the existence of translation invariant abstract upper densities onto the unit interval, whose null sets are precisely the family of finite sets, or the family of sequences whose series of reciprocals converge. We also show that no such density can be atomless. (More generally, these results also hold for a large class of summable ideals.)
