A set $X\subset \mathbb{R}$ is called

niceif for every $\epsilon > 0$ there are a positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \cdots \cup I_k$ and $\sum\limits_{j=1}^k |I_j|^{\epsilon} < \epsilon$.

Prove that there exist sets $X,Y \subset [0,1]$, both of themnice, such that $X+Y = [0,2]$, where $X+Y:=\{x+y\mid x\in X,y\in Y\}.$

This problem was posted some time ago at math.SE.

It's from an Iberoamerican contest for undergraduate students.

The *nice* condition is stronger than Lebesgue measure zero, since there is a $\epsilon$ in the exponent.

No solution was received.
It was suggested the use of continued fractions whose digits are bounded, but I cannot see how this can be useful (I cannot prove or disprove that a set of such continued fractions are nice).

Any suggestion is appreciated. Thanks in advance.