Densities for sets of natural numbers vanishing on a given family

TL;DR

This paper extends the construction of abstract upper densities to all ideals with the Baire property, generalizing previous results for summable ideals and addressing an open problem in the theory of densities.

Contribution

It proves that for any ideal with the Baire property, there exists a 'nice' upper density whose null sets are exactly those in the ideal, broadening the class of ideals covered.

Findings

Constructed densities for all ideals with the Baire property.

Extended previous results from summable ideals to a larger class.

Addressed an open problem posed at a workshop in 2013.

Abstract

Abstract upper densities are monotone and subadditive functions from the power set of positive integers into 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. At the open problem session of the Workshop ``Densities and their application'', held at St. \'{E}tienne in July 2013, G. Grekos asked a question whether there is a ``nice'' abstract upper density, whose the family of null sets is precisely a given ideal of subsets of $\mathbb{N}$, where ``nice'' would mean the properties of the familiar densities consider in number theory. In 2018, M. Di Nasso and R. Jin (Acta Arith. 185 (2018), no. 4) showed that the answer is positive for the summable ideals (for instance, the family of finite sets and the family of sequences whose series of reciprocals converge) when ``nice'' density means translation invariant and rich density (i.e. density which is onto the unit interval). In this paper we extend their result to all ideals with the Baire property.