Approximation and interpolation by large entire cross-sections of second category sets in R^{n+1}

In [M.R. Burke, Large entire cross-sections of second category sets in $\mathbb{R}^{n+1}$, Topology Appl. 154 (2007) 215–240], a model was constructed in which for any everywhere second category set $A\subseteq\mathbb{R}^{n+1}$ there is an entire function $f:\mathbb{R}^n→\mathbb{R}$ which cuts a large section through $A$ in the sense that $\{x∈\mathbb{R}^n:(x,f(x))∈A\}$ is everywhere second category in $\mathbb{R}^n$. 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 $g:\mathbb{R}^n→\mathbb{R}$ is a $C^N$ function, $ε:\mathbb{R}^n→\mathbb{R}$ is a positive continuous function, $T⊆\mathbb{R}^n$ is a closed discrete set, and $G⊆\mathbb{R}^{n+1}$ is a dense $G_δ$ set. Let $A⊆\mathbb{R}^n$ be a countable dense set disjoint from $T$ and for each $x∈A$, let $B_x⊆\mathbb{R}$ be a countable dense set. Then there is a function $f:\mathbb{R}^n→\mathbb{R}$ which is the restriction of an entire function $\mathbb{C}^n→\mathbb{C}$ such that the following properties hold. (a) For all multi-indices $α$ of order at most $N$ and all $x∈\mathbb{R}^n$, $|(D^αf)(x)−(D^αg)(x)|<ε(x)|$, and moreover $(D^αf)(x)=(D^αg)(x)$ when $x∈T$. (b) For each $x∈A$, $f(x)∈B_x$. (c) $\{x∈\mathbb{R}^n:(x,f(x))∈G\}$ is a dense $G_δ$ set in $\mathbb{R}^n$.