Let $f: \Bbb{R}^n \rightarrow \Bbb{R}$ be a nonnegative function that vanishes on a set $\Omega$ that is compact and has positive measure. What is the minimial amount of regularity required of $f$ to guarantee that $\Omega$ contains an open set? I'm interested in classes of the form $C^k$ or $C^{k,\alpha}$ ($k$times continuous differentiability and Hölder spaces, respectively).

By the Whitney extension theorem any closed set of $\mathbb{R}^n$ can be the zeroset of a nonnegative $C^\infty$ function. And, of course, there are closed sets with positive measure and empty interior. 

