Smooth submanifolds intersecting any analytic curve in a discrete set

We construct examples of $C^{\mathrm{\infty }}$ smooth submanifolds in ${\mathbb{C}}^n$ and ${\mathbb{R}}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real $n$-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.