Measure of set of points that are $\epsilon$ away from measure 0 set

Question

Suppose $D \subset \mathbb{R}^n$ is a Lebesgue measure 0 set. Define $U_\epsilon = \{z \in \mathbb{R}^n: distance(z, D) < \epsilon \}$. Can I show that $m(U_{\epsilon}) = O(\epsilon^n)$ as $\epsilon \to 0$? If this is not true how about $O(\epsilon^2)$ and if $D$ is bounded?

Answer 1

This is certainly not true : take $D=\mathbb{Q}^n$, then $U_\epsilon =\mathbb{R}^n$ for every $\epsilon$.

I should add that the boundedness assumption doesn't help : take $D=\mathbb{Q}^n \cap [0,1]^n$ for instance.