Ultrafilters and small sets
by
Jana Flašková
Charles University, Prague, Czech Reublic

There have been several attempts to connect ultrafilters with families of "small" sets. One of them was made by Gryzlov who called a free ultrafilter U on natural numbers a 0-point if for every permutation p of natural numbers there exists U ∈ U such that p[U] has asymptotic density zero. Gryzlov constructed such ultrafilters in ZFC (see [2]).

We strengthened his result in [1] by modifying Gryzlov's definition so that "smallness" of sets is determined by the summable ideal, i.e. the family {A ⊆ N: ∑_{a ∈ A} 1/a < +∞}, which is a proper subideal of the ideal of sets with asymptotic density zero. We say that a free ultrafilter U is a summable ultrafilter if for every one-to-one function f there exists U ∈ U such that f[U] belongs to the summable ideal and we construct summable ultrafilters in ZFC.

[1] Flasková, J., More than a 0-point, submitted to Comment. Math. Univ. Carolin.

[2] Gryzlov, A., Some types of points in N^*, in: Proceedings of the 12th Winter School on Abstract Analysis (Srní, 1984), Rend. Circ. Mat. Palermo (2) 1984, Suppl. No. 6, 137-138.

Date received: January 27, 2006