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Sem. Topologia i T. Mnogości


If I were a rich density

Prelegent: Rafał Filipów

2021-01-13 16:15

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. Etienne 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 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

In my talk I show how to extend their result to all ideals with the
Baire property. This extension was obtained jointly with Jacek Tryba
and the results are published in the paper ``Densities for sets of
natural numbers vanishing on a given family'' (J. of Number Theory 211
(2020), 371-382).