# Asymptotically discrete ultrafilters

Definition 1. A ultrafilter $$\mathcal U$$ on $$\omega$$ is called discrete (resp. nowhere dense) if for any injective map $$f:\omega\to \mathbb R$$ there is a set $$U\in\mathcal U$$ whose image $$f(U)$$ is a discrete (resp. nowhere dense) subspace of the real line $$\mathbb R$$.

It is easy to see that each p-point is a discrete ultrafilter and each discrete ultrafilter is nowhere dense. By a known result of Shelah, there are models of ZFC without nowhere dense ultrafilters.

I am interested in the ZFC-existence of asymptotic counterparts of discrete ultrafilters.

A subset $$D$$ of a metric space $$(X,d)$$ is called asymptotically discrete if for any $$r\in\mathbb N$$ there exists a bounded set $$B\subset X$$ such that $$d(x,y)\ge r$$ for any distinct points $$x,y\in D\setminus B$$.

In particular, a subset $$D$$ of $$\mathbb Z$$ is asymptotically discrete iff for every $$n\in\omega$$ there exists a finite set $$B\subset\mathbb Z$$ such that $$|x-y|\ge n$$ for any distinct points $$x,y\in D\setminus B$$.

Definition 2. A ultrafilter $$\mathcal U$$ on $$\omega$$ is called asymptotically discrete if for every injective map $$f:\omega\to\mathbb Z$$ there exists a set $$U\in\mathcal U$$ whose image $$f(U)$$ is asymptotically discrete in the metric space $$\mathbb Z$$ endowed with the standard Euclidean metric.

In this paper Petrenko and Protasov proved that an ultrafilter on $$\omega$$ is asymptotically discrete if it is a $$p$$-point or a $$q$$-point. They also proved that under CH there are asymptotically discrete ultrafilters which are neither $$p$$-points nor $$q$$-points.

Problem 1. Does an asymptotically discrete ultrafilter exist in ZFC?

Remark 1. If there exists a model of ZFC without asymptotically discrete ultrafilters, then in this model there is no $$p$$-points and no $$q$$-points. The existence of such a model is a well-known open problem in Set Theory. So, if Problem 1 has a simple answer, then this answer is affirmative. Maybe Kunen's OK-ultrafilters are asymptotically discrete?

Remark 2. Any ultrafilter $$\mathcal U$$ containing the filter of subsets of Banach density 1 in $$\omega$$ is not asymptotically discrete (as every subset $$U\in\mathcal U$$ has non-zero Banach density).