It is well known and not difficult to prove that if C ⊆ Z has positive upper Banach density, the set of differences C −C is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever A and B have positive upper Banach density, then A − B is piecewise syndetic. Jin’s result follows trivially from the first statement provided that B has large inter...

Let X be a locally compact Hausdorff space and C0(X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C0(X). They arise from prime ideals of C0(X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c0 can be co...

In the paper we formulate an axiom CPA prism, which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to ω1, while c = ω2: the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number ...

An ultrafilter U on a countable base set B has continuous Tukey reductions if whenever an ultrafilter V is Tukey reducible to U , then every monotone cofinal map f : U → V is continuous with respect to the Cantor topology, when restricted to some cofinal subset of U . We show that the slightly stronger property of having basic Tukey reductions is inherited under Tukey reducibility. It follows t...

We study ultrafilters on ω produced by forcing with the quotient of P(ω) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a Ppoint and that the only non-principal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest squarebracket ...

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: alge...

Our work is a foundational study of the notion of approximation inQ-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (...