Construct a set $A$ for which the lower box dimension of $A$ is less than the upper box dimension of $A$

Question

I'm revising for an exam and would appreciate any answers to this question. Thanks

Answer 1

Consider the set of all numbers of the form $\sum_{k=0}^\infty a_k2^{-k}$ where $a_k\in\{0,1\}$ for all $k$ and additionally $a_k=0$ whenever $\lfloor\log_2 k\rfloor$ is even.