We construct a G δ set G ⊆ ω ω ×2 ω with null vertical sections such that each perfect set P ⊆2 ω meets almost all vertical sections of G in the following sense: we can define from P subsets S of ω of density zero such that whenever the section determined by x ∈ ω ω does not meet P , then x ∈ S for all but finitely many i . This generalizes theorems of Mokobodzki and Brendle (...) et al. 185–199). (shrink)