Approximation and interpolation by large entire cross-sections of second category sets in

In [M.R. Burke, Large entire cross-sections of second category sets in R n + 1 , Topology Appl. 154 (2007) 215–240], a model was constructed in which for any everywhere second category set A ⊆ R n + 1 there is an entire function f : R n → R which cuts a large section through A in the sense that { x ∈ R n : ( x , f ( x ) ) ∈ A } is everywhere second category in R n . Moreover, the function f can be taken so that its derivatives uniformly approximate those of a given C N function g in the sense of a theorem of Hoischen. In the theory of the approximation of C N functions by entire functions, it is often possible to insist that the entire function interpolates the restriction of the C N function to a closed discrete set. In the present paper, we show how to incorporate a closed discrete interpolation set into the above mentioned theorem. When the set being sectioned is sufficiently definable, an absoluteness argument yields a strengthening of the Hoischen theorem in ZFC. We get in particular the following: Suppose g : R n → R is a C N function, ε : R n → R is a positive continuous function, T ⊆ R n is a closed discrete set, and G ⊆ R n + 1 is a dense G δ set. Let A ⊆ R n be a countable dense set disjoint from T and for each x ∈ A , let B x ⊆ R be a countable dense set. Then there is a function f : R n → R which is the restriction of an entire function C n → C such that the following properties hold. (a) For all multi-indices α of order at most N and all x ∈ R n , | ( D α f ) ( x ) − ( D α g ) ( x ) | < ε ( x ) , and moreover ( D α f ) ( x ) = ( D α g ) ( x ) when x ∈ T . (b) For each x ∈ A , f ( x ) ∈ B x . (c) { x ∈ R n : ( x , f ( x ) ) ∈ G } is a dense G δ set in R n .