Density zero slaloms (Q1977484)

Density zero slaloms are functions \(S: \omega \to \mathcal{P} (\omega)\) whose range consists of density zero subsets of \(\omega\). A set \(A \subseteq \omega^\omega\) is density zero if there exists a density zero slalom \(S\) such that for every \(x \in A\) we have \(x(i) \in S(i)\) for all but finitely many \(i\). The main result of the paper is purely combinatorial and asserts the existence of a \(\mathbf{G}_\delta\) set \(G \subseteq \omega^\omega \times 2^\omega\) with null vertical sections which satisfies the following property: whenever \(P \subseteq 2^\omega\) is perfect then \(G_x \cap P\) is dense in \(P\) for all \(x \in \omega^\omega\) but those in a density zero set. This results is effective in the sense that the function assigning to a perfect tree \(T\) the density zero slalom witnessing that the set \(\{x : G_x \cap [T]\) is not dense in \([T]\}\) is density zero can be chosen to be continuous. After proving the existence of \(G\), the author applies it to generalize a theorem of \textit{G. Mokobodzki} [Séminaire de Probabilités XII, Université de Strasbourg 1976/77, Lect. Notes Math. 649, 491-508 (1978; Zbl 0401.28002)] by proving that if \(A \subseteq 2^\omega \times 2^\omega\) is an analytic set with uncountable vertical sections except on a null set then for all \(y\) but those in a density zero set \(A_x \cap G_y\) is uncountable for every \(x\) but those in a null set. This theorem is further generalized by replacing the ideal of null sets with any finite product of the ideals of null and meager sets. All results mentioned so far are proved within ZFC, but the motivation for the whole paper appears to lie in the study of forcing extensions of models of ZFC. Indeed density zero slaloms are connected to the existence of perfect sets of random reals, as a further application of the main result shows. In fact, the author generalizes a result of \textit{J. Brendle, H. Judah} and \textit{S. Shelah} [Ann. Pure Appl. Logic 58, 185-199 (1992; Zbl 0790.03054)] by proving that if \(V\) is a transitive model of ZFC such that the set of random reals over \(V\) contains a perfect set then \(V \cap \omega^\omega\) is a density zero set; in turn the latter property implies that either \(V \cap \omega^\omega\) is contained in a \(\sigma\)-compact set or \(V \cap 2^\omega\) is contained in a null \(\mathbf{F}_\sigma\). None of the two implications can be reversed. The last section studies briefly other kind of slaloms and relates them to other kinds of forcings.