Abstract densities and ideals of sets

TL;DR

This paper explores abstract upper densities on sets of positive integers, proving the existence of translation invariant densities with specific null sets and showing such densities cannot be atomless, extending to summable ideals.

Contribution

It answers a question by G. Grekos by constructing translation invariant densities with prescribed null sets and demonstrates the impossibility of atomless densities with these properties.

Findings

Existence of translation invariant densities with null sets as finite sets or convergent reciprocal series.

No translation invariant density with these null sets can be atomless.

Results extend to a broad class of summable ideals.

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.)