CIIM 2010 Problem 3

A set $X\subset \mathbb{R}$ has dimension zero if, for any $\epsilon > 0$ there exists a positive integer $k$ and intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \cdots \cup I_k$ with $\sum_{j=1}^k |I_j|^{\epsilon} < \epsilon$ .

Prove that there exist sets $X,Y \subset [0,1]$ both of dimension zero, such that $X+Y = [0,2].$