I am trying to find two subsets $E,F$ of $\mathbb{R}$ that satisfy $\underline{\text{dim}}_{B}(E\cup F)>\max\{\underline{\text{dim}}_{B}(E),\underline{\text{dim}}_{B}(F)\}$, where $\underline{\text{dim}}_{B}$ is the lower box-counting dimension.
I've been given the hint: Let $k_{n}=10^{n}$ and adapt the Cantor set construction by deleting at the $k$-th stage, the middle $1/3$ of intervals if $k_{2n}<k\leq k_{2n+1}$ and the middle $3/5$ of intervals if $k_{2n-1}<k\leq k_{2n}$. But I'm not even sure what it means when it says "if $k_{2n}<k\leq k_{2n+1}$".
