Let me begin by quoting a well-known result of Christian Baer (see [here][1]). The result goes as follows: 

Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ and generalized Dirac operator $D$. Let $h$ be a smooth endomorphism field for $S$ and $s$ be a non-zero solution of $(D + h)s = 0$. Then the nodal set $N_h$ of $s$ has Hausdorff dimension $(n - 2)$ at most.

The proof actually gives more information, I have only mentioned the salient parts that pertain to my questions, which are as follows:

(a) Are sufficient conditions known so that the $(n - 2)$-dimensional Hausdorff measure of $N_h$ is non-zero?

(b) Are there (upper and lower) estimates on the $(n - 2)$-dimensional  Hausdorff measure of $N_h$ in terms of $h$?

This is mainly a reference request.





[1]:http://arxiv.org/pdf/dg-ga/9707008v1.pdf