I am stuck in the following question:
A set $X\subset\mathbb{R}$ has dimension zero if, for all $\epsilon>0$, there exists an integer $k$ and limited intervals $I_1,\cdots,I_k$ such that $X\subset I_1\cup\cdots\cup I_k$ and $\sum_{j=1}^k |I_j|^{\epsilon}<\epsilon$.
Show that exists sets $X,Y\subset [0,1]$, both with dimension zero such that $X+Y=[0,2]$, where $X+Y:=\{x+y\:|\:\: x\in X, y\in Y\}$. PS: $|I|$ denotes the lenght of the interval $I$.
I know that we must have $\{0,1\}\subset X\cap Y$. But I can't quite figure out whats the reasoning behing the definition of dimension zero and how to find such sets.
It means Hausdorff dimension less than $d$ for all $d>0$: for all $d,\epsilon > 0$ there are coverings of the set by $\cup I_j$, with $\sum |I_j|^d < \epsilon$. It is enough to meet that condition for all pairs $(d,\epsilon)$ with $d = \epsilon$, which is the slightly confusing form of the definition in the problem.
The usual examples of $X+X=2$ with Cantor sets have measure $0$ but are positive dimensional, and I think that finding examples in $0$-dimensional sets is a well known problem solved or posed by Erdos in the 1940's. Of course the first thing to try is thinner Cantor-like sets but it is not clear how to make that work.