Let $f: \Bbb{R}^n \rightarrow \Bbb{R}$ be a non-negative 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).

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One first thought is that for $n=1$ you probably need $\alpha\geq 1/2$, although I haven't thought through this in detail – Yemon Choi Jun 25 '11 at 20:34
should be "to this site and take it over" – Will Jagy Jun 25 '11 at 20:36
@Jagy: point taken. – Viktor Bundle Jun 25 '11 at 20:57

By the Whitney extension theorem any closed set of $\mathbb{R}^n$ can be the zero-set of a non-negative $C^\infty$ function. And, of course, there are closed sets with positive measure and empty interior.
or more directly, a smooth non-negative function vanishing on a prescribed closed $\Omega$ can be constructed by means of a partition of unity. – Pietro Majer Jun 25 '11 at 20:49