Zygmund graphs are thin for doubling measures

The Zygmund functions form an intermediate class between Lipschitz and Hölder functions; their second order divided differences are uniformly bounded. It is well known that for $d\ge 1$ the graph of any Lipschitz function $f:{\mathbb{R}}^d\to \mathbb{R}$ is thin for doubling measures, and we extend this result to the Zygmund class.